lecture notes sta 2101 algebra for statistics and finance · 2017-09-19 · time is precious, but...
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Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 1
Lecture Notes
STA 2101 ALGEBRA FOR STATISTICS AND FINANCE Pre-Requisites: None
Course Purpose By the end of this course, the student will have a good understanding of
algebraic notations, and gained knowledge on how to reason symbolically. Emphasis will be
placed on the study of functions, and their graphs, inequalities, and linear, quadratic, piece-wise defined,
rational, polynomial, exponential, and logarithmic functions.
Learning Objectives By the end of this course the student should be able to;
i) Solve linear, quadratic, and other nonlinear equations and be able to graph the solutions, using pencil and
paper as well as the graphics calculator.
ii) Recognize equations and graphs involving the conic sections.
iii) Recognize graphs and properties of functions and their inverses and interpret data involving graphs.
iv) Perform operations and procedures on polynomials, rational expressions, and numbers (real and
complex).
v) Use exponential and logarithmic functions in practical applications.
vi) Solve systems of equations using two and three variables by various methods, including the graphic
calculator.
vii) Use arithmetic and geometric sequences and series to find sums and products and to solve practical
application problems.
viii) Perform factorial notations and the binomial expansion.
ix) Apply permutations and combinations to fundamental principles of probability.
Course Description
Graphing parabolas, circles, ellipses, and hyperbolas; relations and functions; graph- ing functions; combining
functions; inverse functions; direct an inverse variations; solving problems whose mathematical models are
polynomial, rational, exponential and logarithmic functions; finding zeros of polynomial and rational
functions; solv- ing systems of linear and nonlinear equations and inequalities with applications for each;
matrices and determinants; systems of nonlinear equations; binomial expan- sions; arithmetic and geometric
sequences and series; and counting techniques.
Course Text Books and Reference Text Books
1) Schay G., A Concise Introduction to Linear Algebra, Springer, ISBN: 9780817683252, 2012.
2) Strang G., Introduction to Linear Algebra, SIAM, ISBN: 9780961408893, 2003.
3) Hamilton A.G., Linear Algebra: An Introduction. Cambridge University Press, ISBN:
052132517X, 1989.
4) Andrilli S., Elementary Linear Algebra, Gulf Professional Publishing, ISBN: 9780120586219, 2003
5) Minc H., Introduction to Linear Algebra, Courier Dover Publications, ISBN: 9780486656953, 1988.
6) Lang S., Linear Algebra, Springer, ISBN: 9780387964126, 1987. 1573-8795.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 2
VARIATIONS IN ALGEBRA Direct Variation If you know the retail price of one taifa laptop at JKUAT, then you can determine the total
revenue after selling 100, 1,000 or 15,000laptops since the total revenue is a constant multiple of
the retail price of one laptop. When two quantities y and x have a constant ratio k, they are said
to have direct variation and we write kxyxy then . where 0k
The constant k is called the constant of
Variation The equation kxy represents
direct variation between x and y, and y is
said to vary directly with x.
The graph of a direct variation equation
kxy is a line with slope k and y-intercept
0. The family of direct variation graphs
consists of lines through the origin, such as
those shown. Example 1 If y varies directly as x and y=20 when x=5 Write an equation that relates x and y
hence find the value of y when x=12.
Solution
Because x and y vary directly, then kxyxy Put x=5 and y=20 then
4 )5(20 kk Now when 48124, 12 yx
Example 2 Write and graph a direct
variation equation that has (-4, 8) as a
solution.
Solution
Use the given values of x and y to find the
constant of variation. aaxy 48
2a . Substituting -2 for a in axy
gives the direct variation equation xy 2 .
Its graph is shown.
Example 3 Hailstones form when strong updrafts support ice particles high in clouds, where
water droplets freeze onto the particles. The diagram shows a hailstone at two different times
during its formation.
a) Write an equation that gives the
hailstone’s diameter d (in inches) after t
minutes if you assume the diameter
varies directly with the time the
hailstone takes to form.
b) Using your equation from part (a),
predict the diameter of the hailstone after
20 minutes
c) Suppose that a hailstone forming in a
cloud has a radius of 0.6 inch. Predict
how long it has been forming.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 3
Solution
a) Use the given values of t and d to find the constant of variation. aatd 1275.0
0625.0a . An equation that relates t and d is td 0625.0 .
b) After t=20 minutes, the predicted diameter of the hailstone is 25.1)20(0625.0 d inches.
c) Using the equation that relates t and d ie td 0625.0 put 6.0d to get
6.90625.06.0 tt minutes,
Because the direct variation equation kxy can be written as kx
y , a set of data pairs (x, y)
shows direct variation if the ratio of y to x is constant.
Example 4 Great white sharks have triangular teeth. The table below gives the length of a side
of a tooth and the body length for each of six great white sharks.
a) Does tooth length and body length show direct variation? If yes, write an equation that relates
the quantities.
Tooth length, t (cm) 1.8 2.4 2.9 3.6 4.7 5.8
Body length, b(cm) 215 290 350 430 565 695
b) The respective body masses m (in kilograms) of the great white sharks are; 80, 220, 375, 730,
1690, and 3195. Tell whether tooth length and body mass show direct variation. If so, write
an equation that relates the quantities
Solution
Find the ratio of the body length b to the tooth length t for each shark.
a) 44.48.1
80 , 91.7
4.2
220 , 3.129
9.2
375 , 8.202
6.3
730 , 6.359
7.4
1690 ,
1208.5
695
Because the ratios are approximately equal, the data show direct variation. An equation
relating tooth length and body length is tbt
b120120
b) Repeat the same procedure.
448.1
80 , 92
4.2
220 , 129
9.2
375 , 203
6.3
730 , 360
7.4
1690 , 551
8.5
3195
Because the ratios differ significantly, the data does not show direct variation.
Exercise
1) Write and graph a direct variation equation that has the given ordered pair as a solution.
(2, 6). (-3, 12). (6, -21). (4, 10). (-5, -1). (24, -8). (4
3, -4). (12.5, 5).
2) The variables x and y vary directly. Write an equation that relates x and y. Then find y
when x=12.
X=4, y=8 X=-3, y=-5 X=35, y=-7 X=-18, y=4 X=-4.8, y=-1.6, X=2
3, y=-10
3) Which equation is a direct variation equation that has (3,18) as a solution? 𝑦 = 2𝑥2,
𝑦 = 1
6𝑥, 𝑦 = 6𝑥 , 𝑦 = 4𝑥 + 6
4) Does the given equation represents direct variation? If so, give the constant of variation?
𝑦 = 8𝑥, 𝑦 = 4 − 3𝑥, 3𝑦 − 7 = 10𝑥 , 2𝑦 − 5𝑥 = 0, 5𝑦 = −4𝑥 , 6𝑦 = 𝑥
5) The variables x and y vary directly. Write an equation that relates x and y. Then find x
when y=24.
X=5, y=15 X=-6, y=8 X=-18, y=-2 X=-12, y=84 X=-20
3, y=-15
8 X=-0.5, y=3.6,
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 4
6) Does the data in the table show direct variation? If so, write an equation relating x and y.
X 3 6 9 12 15
Y -1 -2 -3 -4 -5
X 1 2 3 4 5
Y 7 9 11 13 15
X -8 -4 4 8 12
y 8 4 -4 -8 -12
X -5 -4 -3 -2 -1
y 20 16 12 8 4
7) Let (x1,y1) be a solution, other than (0, 0), of a direct variation equation. Write a second
direct variation equation whose graph is perpendicular to the graph of the first equation.
8) Let (x1,y1) and (x2 , y2) be any two distinct solutions of a direct variation equation. Show
that 𝑥2
𝑥1=
𝑦2
𝑦1
9) The time t it takes a diver to ascend safely to the surface varies directly with the depth d.
It takes a minimum of 0.75 minute for a safe ascent from a depth of 45 feet. Write an
equation that relates d and t. Then predict the minimum time for a safe ascent from a
depth of 100 feet.
10) Hail 0.5 inch deep and weighing 1800 pounds covers a roof. The hail’s weight w varies
directly with its depth d. Write an equation that relates d and w. Then predict the weight
on the roof of hail that is 1.75 inches deep.
11) Your weight M on Mars varies directly with your weight E on Earth. If you weigh 116
pounds on Earth, you would weigh 44 pounds on Mars. Which equation relates E and M?
𝑀 = 𝐸 − 72 44𝑀 = 116𝐸 𝑀 = 28
11𝐸 𝑀 = 11
28𝐸
12) The ordered pairs (4.5, 23), (7.8, 40), and (16.0, 82) are in the form (s, t) where t
represents the time (in seconds) needed to download an
Internet file of size s (in megabytes). Tell whether the data show direct variation. If so,
write an equation that relates s and t.
13) Each year, gray whales migrate from Mexico’s Baja Peninsula to feeding grounds near
Alaska. A whale may travel 6000 miles at an average rate of 75 miles per day.
a) Write an equation that gives the distance d1 traveled in t days of migration.
b) Write an equation that gives the distance d2 that remains to be traveled after t days of
migration.
c) Tell whether the equations from parts a) and b) represent direct variation. Explain your
answers.
14) At a jewelry store, the price p of a gold necklace varies directly with its length l. Also, the
weight w of a necklace varies directly with its length. Show that the price of a necklace
varies directly with its weight.
Inverse and Joint Variation
In the previous section you learned that two variables x and y show direct variation if y = kx for
some nonzero constant k. Another type of variation is called inverse variation. Two variables x
and y show inverse variation if increase in one variable leads to decrease in the other variable
and we write x
k
xyy 1
. where 0k is called the constant of variation, and y is said to
vary inversely with x. The equation for inverse variation can be rewritten as xy = k. This tells
you that a set of data pairs (x, y) shows inverse variation if the products xy are constant or
approximately constant.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 5
Example 1 The variables x and y vary inversely, and y = 8 when x = 3. Write an equation that
relates x and y hence find y when 𝑥 = −4.
Solution
Use the given values of x and y to find the constant of variation x
k
xyy 1
or xyk .
Thus 242483 xyk The inverse variation equation is xy 24 Now When 𝑥 = −4.,
the value of y is: 6424 y
Example 2 The speed of the current in a whirlpool varies inversely with the distance from the
whirlpool’s center. The Lofoten Maelstrom is a whirlpool located off the coast of Norway. At a
distance of 3 kilometers (3000 meters) from the center, the speed of the current is about 0.1
meter per second. Describe the change in the speed of the current as you move closer to the
whirlpool’s center.
Solution
First write an inverse variation model relating distance from center d and speed s
d
k
dss 1
or sdk . Thus 30030030001.0 sdk
The inverse variation equation is ds 300 The table shows some speeds for different values of d.
.d 2000 1500 500 250 50
s 0.15 0.2 0.6 1.2 6
From the table you can see that the speed of the current increases as you move closer to the
whirlpool’s center.
Example 3 The table compares the wing
flapping rate r (in beats per second) to the
wing length l (in centimeters) for several
birds. Do these data show inverse variation?
If so, find a model for the relationship
between r and l.
Solution
Each product rl is approximately equal to 117. For instance, (3.6)(32.5) = 117 and (5.0)(23.5) =
117.5. So, the data do show inverse variation. A model for the relationship between wing
flapping rate and wing length is lr 117
Joint variation
Joint Variation occurs when a quantity varies directly as the product of two or more other
quantities. For instance, if 𝑧 = 𝑘𝑥𝑦 where k ≠ 0, then z varies jointly with x and y. Other types
of variation are also possible, for instance
i) z varies directly as x and inversely with y.. So y
x
y
x kzz
ii) z varies directly as y and inversely with the square of x.. So 22 x
kyz
x
yz
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 6
iii) z varies directly as the square of x and inversely with the square root of y.. So
22
y
kxz
y
xz
Example 1 The law of universal gravitation states that the gravitational force F (in newtons)
between two objects varies jointly with their masses m1 and m2 (in kilograms) and inversely
with the square of the distance d (in meters)
between the two objects. The constant of
variation is denoted by G and is called the
universal gravitational constant. Write an
equation for the law of universal gravitation
hence estimate the universal gravitational
constant. Use the Earth and sun facts given
at the right
Solution
2
21
d
mGmF Substitute the given values and solve for G
)1067.6 G )1029.5(G )1050.1(
)1099.1)(1098.5(1053.3 1132
211
302422
G
The universal gravitational constant is about 2211 /1067.6 KgNM
Exercise
1) Tell whether x and y show direct variation, inverse variation, or neither
𝑥𝑦 = 0.25, 𝑥
𝑦= 5, 𝑦 = 𝑥 − 3 , 𝑥 = 7
𝑦, 𝑦
𝑥= 12 , 1
2𝑥𝑦 = 9, 2𝑥 + 𝑦 = 4
2) The variables x and y vary inversely. Use the given values to write an equation relating x and
y. Then find y when x = 2. (a) X=5, y=-2 (b) X=4, y=8 (c) X=7, y=1 (d) X=0.5, y=10 (e)
𝑋 = 2
3-, y=-6, (f) 𝑋 = 3
4-, y=-3
8,
3) Determine whether x and y show direct variation, inverse variation, or neither
x y b) x y © x y d) x y
1.5 20 31 217 3 36 1.6 40
2.5 12 20 140 5 50 4 16
4 7.5 17 119 7 105 5 12.8
5 6 12 84 16 48 20 3.2
4) Write an equation for the given relationship.
a) x varies inversely with y and directly with z.
b) y varies jointly with z and the square root of x.
c) w varies inversely with x and jointly with y and z.
5) The variable z varies jointly with x and y. Use the given values to write an equation relating
x, y, and z. Then find z when x =-4 and y = 7. (a) X=3, y=8. Z=6 (b) X=-12, y=4, z=2 (c)
X=1, y=1
3, z=5 (d) X=-6, y=3, z=2
3 (e) 𝑋 = 5
6-, y=0.3, z=8 (f) 𝑋 = 3
8,-, y=-16
17,, z=3
2,
6) Tell whether x varies jointly with y and z. (a) 𝑥 = 15𝑦𝑧, (b) 𝑥
𝑧= 0.5𝑦, (c) 𝑥𝑦 = 4𝑧 , (d)
𝑥 = 𝑦𝑧
2, (e) x= 3𝑧
𝑦, (f) 2𝑦𝑧 = 7𝑥, (g) 𝑥
𝑦= 17𝑧, (h) 5𝑥 = 4𝑦𝑧
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 7
7) The amount A (in dollars) you pay for grapes varies directly with the amount P (in pounds)
that you buy. Suppose you buy 1.5 pounds for $2.25. Write a linear model that gives A as a
function of P.
8) The force F needed to loosen a bolt with a wrench varies inversely with the length l of the
handle. Write an equation relating F and l, given that 250 pounds of force must be exerted to
loosen a bolt when using a wrench with a handle 6 inches long. How much force must be
exerted when using a wrench with a handle 24 inches long?
9) On some tubes of caulking, the diameter of the circular nozzle opening can be adjusted to
produce lines of varying thickness. The table shows the length l of caulking obtained from a
tube when the nozzle opening has diameter d and cross-sectional area A.
a) Determine whether l varies inversely with d. If so, write an equation relating l and d.
b) Determine whether l varies inversely with A. If so, write an equation relating l and A.
c) Find the length of caulking you get
from a tube whose nozzle opening
has a diameter of 3 4 inch.
D(in.) 81 4
1 83
21
A(in.2) 256
64
2569
16
I(in.) 1440 360 160 90
10) The number of admission applications received by a college was 1152 in 1990 and increased
5% per year until 1998. Write a model giving the number A of applications t years after
1990. Hence graph the model. Use the graph to estimate the year in which there were 1400
applications.
11) A star’s diameter D (as a multiple of the sun’s diameter) varies directly with the square root
of the star’s luminosity L (as a multiple of the sun’s luminosity) and inversely with the
square of the star’s temperature T (in kelvins).
a) Write an equation relating D, L, T, and a constant k.
b) The luminosity of Polaris is 10,000 times the luminosity of the sun. The surface
temperature of Polaris is about 5800 kelvins. Using k = 33,640,000, find how the
diameter of Polaris compares with the diameter of the sun.
c) The sun’s diameter is 1,390,000 kilometers. What is the diameter of Polaris?
12) The work W (in joules) done when lifting an object varies jointly with the mass m (in
kilograms) of the object and the height h (in meters) that the object is lifted. The work done
when a 120 kilogram object is lifted 1.8 meters is 2116.8 joules. Write an equation that
relates W, m, and h. How much work is done when lifting a 100 kilogram object 1.5 meters?
13) The intensity I of a sound (in watts per square meter) varies inversely with the square of the
distance d (in meters) from the sound’s source. At a distance of 1 meter from the stage, the
intensity of the sound at a rock concert is about 10 watts per square meter. Write an equation
relating I and d. If you are sitting 15 meters back from the stage, what is the intensity of the
sound you hear?
14) The heat loss h (in watts) through a single-pane glass window varies jointly with the
window’s area A (in square meters) and the difference between the inside and outside
temperatures d (in kelvins).
a) Write an equation relating h, A, d, and a constant k.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 8
b) A single-pane window with an area of 1 square meter and a temperature difference of 1
kelvin has a heat loss of 5.7 watts. What is the heat loss through a single-pane window
with an area of 2.5 square meters and a temperature difference of 20 kelvins?
15) The area of a trapezoid varies jointly with the height and the sum of the lengths of the bases.
When the sum of the lengths of the bases is 18 inches and the height is 4 inches, the area is
36 square inches. Find a formula for the area of a trapezoid.
16) The load P (in pounds) that can be safely supported by a horizontal beam varies jointly with
the width W (in feet) of the beam and the square of its depth D (in feet), and inversely with
its length L (in feet). a. How does P change when the width and length of the beam are
doubled? b. How does P change when the width and depth of the beam are doubled? c. How
does P change when all three dimensions are doubled? Describe several ways a beam can be
modified if the safe load it is required to support is increased by a factor of 4.
17) Suppose x varies inversely with y and y varies inversely with z. How does x vary with z?
Justify your answer algebraically
18) Ohm’s law states that the resistance R (in ohms) of a conductor varies directly with the
potential difference V (in volts) between two points and inversely with the current I (in
amperes). The constant of variation is 1. What is the resistance of a light bulb if there is a
current of 0.80 ampere when the potential difference across the bulb is 120 volts?
LINEAR INEQUALITIES Solving Linear Inequalities with one Variable Inequalities have properties that are similar to those of equations, but the properties differ in
some important ways. Inequalities such as 𝑥 ≤ 1 and 2𝑛 − 3 > 9 are examples of linear
inequalities in one variable. A solution of an inequality in one variable is a value of the variable
that makes the inequality true. For instance,-2, 0, 0.872, and 1 are some of the many solutions of
𝑥 ≤ 1.
In the activity you may have discovered some of the following properties of inequalities. You
can use these properties to solve an inequality because each transformation produces a new
inequality having the same solutions as the original.
Transformations that Produce Equivalent Inequalities
Add or subtract the same number to both sides.
Multiply or divide both sides by the same positive number.
Multiply or divide both sides by the same negative number and reverse the inequality.
The graph of an inequality in one variable consists of all points on a real number line that
correspond to solutions of the inequality. To graph an inequality in one variable, use an open dot
for < or > and a solid dot for ≤ or ≥. For example, the graphs of
𝑥 < 3 and 𝑥 ≥ −2 are shown below.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 9
Example 1 Solve 5𝑦 − 8 < 12. Solution
5𝑦 − 8 < 12 ⇛ 5𝑦 < 20 ⇛ 𝑦 < 4 The solutions are all real numbers less than 4, as
shown in the graph below.
Example 2 Solve 2𝑥 + 1 ≤ 6𝑥 − 1.
Solution
2𝑥 + 1 ≤ 6𝑥 − 1 original inequality.
−4𝑥 + 1 ≤ −1 Subtract 6x from each side
−4𝑥 ≤ −2 Subtract 1 from each side.
𝑥 ≥ 0.5 Divide each side by -4 and reverse the inequality.
The solutions are all real numbers greater than or equal to 0.5
Example 3
The weight w (in pounds) of an Icelandic saithe is given by 𝑤 = 10.4𝑡 − 2.2 where t is the age
of the fish in years. Describe the ages of a group of Icelandic saithe that weigh up to 29 pounds.
Solution
𝑤 ≤ 29 Weights are at most 29 pounds ⇛ 10.4𝑡 − 2.2 < 29 ie put 𝑤 = 10.4𝑡 − 2.2
⟹ 10.4𝑡 = 31.2 Add 2.2 to each side. ∴ 𝑡 ≤ 3 Divide both side by 10.4. The ages are less
than or equal to 3 years.
Example 4 If 𝑥 ∈ {−3, −2, −1, 0, 1, 2, 3}.find the solution set of each of the following
a) 𝑥 + 2 < 1 b) 2𝑥 − 1 < 4 c) 3 − 5𝑥 < −1 d) − 6 ≥ 2𝑥 − 4 e) 14 − 2𝑥 ≤ 6 Solution
a) 𝑥 + 2 < 1 ⟹ 𝑥 < −1 Subtracting 2 from both sides. Therefore the solution set is {−3, −2} b) 2𝑥 − 1 < 4 ⟹ 2𝑥 < 5 ⟹ 𝑥 < 2.5 Therefore the solution set is {−3, −2, −1, 0, 1, 2} c) 3 − 5𝑥 < −1 ⟹ −5𝑥 < −4 ⟹ 𝑥 > 0.8 Therefore the solution set is {1, 2, 3}. d) −6 ≥ 2𝑥 − 4 ⟹ 10 ≥ 2𝑥 ⟹ 5 ≥ 𝑥 or 𝑥 ≤ 5 Thus the solution set
is{−3, −2, −1, 0, 1, 2, 3} e) 14 − 2𝑥 ≤ 6 ⟹ −2𝑥 ≤ −8 or 𝑥 ≥ 4 Thus the solution set is{} or 𝜙
Solving Compound Inequalities
A Compound Inequality is two simple inequalities joined by “and” or “or.” Here are two
examples
−2 ≤ 𝑥 < 1
All real numbers that are greater than or
equal to -2 and less than 1.
𝑥 < −1 and 𝑥 ≥ 2
All real numbers that are less than -1 or
greater than or equal to 2.
Example 1 Solve −2 ≤ 3𝑡 − 8 ≤ 10.
Solution
To solve, you must isolate the variable between the two inequality signs.
−2 ≤ 3𝑡 − 8 ≤ 10 Write original inequality. ⟹ 6 ≤ 3𝑡 ≤ 18 Add 8 to each expression.
∴ 2 ≤ 𝑡 ≤ 6 Divide each expression by 3.
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Because t is between 2 and 6, inclusive, the solutions are all real numbers greater than or equal to
2 and less than or equal to 6. The graph is shown below.
Example 2 Solve 2𝑥 + 3 < 5 or 4𝑥 − 7 > 9.
Solution
A solution of this compound inequality is a solution of either of its simple parts, so you should
solve each part separately. 2𝑥 + 3 < 5 ⟹ 𝑥 < 1 and 4𝑥 − 7 > 9 ⟹ 𝑥 > 4 thus the
solutions are all real numbers less than 1 or greater than 4. The graph is shown below.
Example 3 You have added enough antifreeze to your car’s cooling system to lower the
freezing point to º35°C and raise the boiling point to 125°C. The coolant will remain a liquid as
long as the temperature C (in degrees Celsius) satisfies the inequality −35 < 𝐶 < 125. Write
the inequality in degrees Fahrenheit.
Solution
Let F represent the temperature in degrees Fahrenheit, and use the formula 𝐶 =5
9(𝐹 − 32).
−35 <5
9(𝐹 − 32) < 125 ⟹ −63 < 𝐹 − 32 < 225 ⟹ −31 < 𝐹 < 257
The coolant will remain a liquid as long as the temperature stays between -31°F and 257°F.
Example 4 You are a state patrol officer who is assigned to work traffic enforcement on a
highway. The posted minimum speed on the highway is 45 miles per hour and the posted
maximum speed is 65 miles per hour. You need to detect vehicles that are traveling outside the
posted speed limits.
a) Write these conditions as a compound inequality.
b) Rewrite the conditions in kilometers per hour.
Solution
a) Let m represent the vehicle speeds in miles per hour. The speeds that you need to detect are
given by: 𝑚 < 45 or 𝑚 > 65
b) Let k be the vehicle speeds in kilometers per hour. The relationship between miles per hour
and kilometers per hour is given by the formula m≈ 0.621k. You can rewrite the conditions in
kilometers per hour by substituting 0.621k for m in each inequality and then solving for k. ie
0.621𝑘 < 45 or 0.621𝑘 > 65 ⟹ 𝑘 < 72.5 or 𝑘 > 105 You need to detect vehicles whose speeds are less than 72.5 kilometers per hour or greater than
105 kilometers per hour.
Linear Inequalities in Two Variables A Linear Inequality in two variables is an inequality that can be written in one of the following
forms: 𝐴𝑥 + 𝐵𝑦 < 𝐶, 𝐴𝑥 + 𝐵𝑦 ≤ 𝐶, 𝐴𝑥 + 𝐵𝑦 > 𝐶, 𝐴𝑥 + 𝐵𝑦 ≥ 𝐶. An ordered pair (x ,y)
is a of a linear inequality if the inequality is true when the values of x and y are substituted into
the inequality. Eg (-6, 2) is a solution of 𝑦 ≥ 3𝑥 − 9 because 2 ≥ 3(−6) − 9 is a true statement.
Example 1 Is the given ordered pair is a solution of 2𝑥 + 3𝑦 ≥ 5? a) (0, 1) b (4, -1) c (2, 1)
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Solution
Ordered Pair Substitute Conclusion
a. (0, 1) 2 × 0 + 3 × 1 = 3 ≱ 5 (0, 1) is not a solution.
b. (4, -1) 2 × 4 + 3 × −1 = 5 ≥ 5 (4, º1) is a solution.
c. (2, 1) 2( 2) + 3(1) = 7 ≥ 5 (2, 1) is a solution.
Graphing a Linear Inequality
The graph of a linear inequality in two variables is the graph of all solutions of the inequality.
The boundary line of the inequality divides the coordinate plane into two a shaded region which
contains the points that are solutions of the inequality, and an unshaded region which contains
the points that are not. To graph a linear inequality, follow these steps:
Graph the boundary line of the inequality. Use a dashed line for < or > and a solid line for ≤
or ≥.
To decide which side of the boundary line to shade, test a point not on the boundary line to
see whether it is a solution of the inequality. Then shade the appropriate half-plane
Example 2 Graph a) ) 𝑦 < −2and b) 𝑥 ≤ 1 in a coordinate plane
Solution
a) Graph the boundary line 𝑦 = −2. Use a dashed line because 𝑦 < −2. Test the point (0, 0).
Because (0, 0) is not a solution of the inequality, shade the half-plane below the line.
b) Graph the boundary line 𝑥 = 1. Use a solid line because 𝑥 ≤ 1. Test the point (0, 0). Because
(0, 0) is a solution of the inequality, shade the half-plane to the left of the line.
Example 3 Graph a) 𝑦 < 2𝑥 and b) 2𝑥 − 5𝑦 ≥ 10 in a coordinate plane
Solution
a) Graph the boundary line 𝑦 = 2𝑥 Use a dashed line because 𝑦 < 2𝑥 Test the point (1, 1).
Because (1, 1) is a solution of the inequality, shade the half-plane below the line.
b) Graph the boundary line 2𝑥 − 5𝑦 = 10 Use a solid line because 2𝑥 − 5𝑦 ≥ 10 Test the
point (0, 0). Because (0, 0) is not a solution of the inequality, shade the half-plane below the
line
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Example 4 You have relatives living in both the United States and Mexico. You are given a
prepaid phone card worth $50. Calls within the continental United States cost $0.16 per minute
and calls to Mexico cost $0.44 per minute.
a) Write a linear inequality in two variables to represent the number of minutes you can use for
calls within the United States and for calls to Mexico.
b) Graph the inequality & discuss 3 possible solutions in the context of the real-life situation.
Solution
a) Let x and y be the number of minutes you can use for calls within the United States and for
calls to Mexico respectively then 0.16𝑥 + 0.44𝑦 ≤ 50
b) Graph the boundary line 0.16𝑥 + 0.44𝑦 = 50. Use a solid line because 0.16𝑥 + 0.44𝑦 ≤ 50
Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane below
the line. Finally, because x and y cannot be negative, restrict the graph to points in the first
quadrant. Possible solutions are points within the shaded region shown.
Graphing and Solving Systems of Linear Inequalities The following is a Systems of Linear Inequalities in two variables.
𝑥 + 𝑦 ≤ 6 Inequality 1 2𝑥 − 𝑦 > 4 Inequality 2
A solution of a system of linear inequalities is an ordered pair that is a solution of each inequality
in the system. For example, (3, -1) is a solution of the system above. The graph of a system of
linear inequalities is the graph of all solutions of the system.
Investigating Graphs of Systems of Inequalities
The coordinate plane shows the four regions
determined by the lines 23 yx and
12 yx . Use the labeled points to help
you match each region with one of the
systems of inequalities
a) 12
23
yx
yx c)
12
23
yx
yx
b) 12
23
yx
yx d)
12
23
yx
yx
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Conclusion; A system of linear inequalities defines a region in a plane. Here is a method for
graphing the region.
Graphing a System of Linear Inequalities
To graph a system of linear inequalities, do the following for each inequality in the system:
• Graph the line that corresponds to the inequality. Use a dashed line for an inequality with <
or > and a solid line for an inequality with ≤ or ≥.
• Lightly shade the half-plane that is the graph of the inequality. Colored pencils may help you
distinguish the different half-planes.
The graph of the system is the region common to all of the half-planes. If you used colored
pencils, it is the region that has been shaded with every color.
Example 1 Graph the system. 2 Inequality2 and 1 Inequality13 xyxy
Solution
Begin by graphing each linear inequality. Use a different color for each half-plane. For instance,
you can use red for Inequality 1 and blue for Inequality 2. The graph of the system is the region
that is shaded purple
Example 2 Graph the system. 2434and0,0 yxyx
Solution
Inequality 1 and Inequality 2 restrict the solutions to the first quadrant. Inequality 3 is the half-
plane that lies on and below the line 2434 yx . The graph of the system of inequalities is the
triangular region shown below
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One can use a system of linear inequalities to describe a real-life situation, as shown in the
following example
Example 3 A person’s theoretical maximum heart rate is 220 − 𝑥 where x is the person’s age
in years (20 ≤ x≤ 65). When a person exercises, it is recommended that the person strive for a
heart rate that is at least 70% of the maximum and at most 85% of the maximum.
a) You are making a poster for health class. Write and graph a system of linear inequalities that
describes the information given above.
b) A 40-year-old person has a heart rate of 150 (heartbeats per minute) when exercising. Is the
person’s heart rate in the target zone?
Solution
a) Let y represent the person’s heart rate. From the given information, you can write the
following four inequalities
65and20 xx Person’s age must be at least 20 and at most 65
)220(8.0and)220(7.0 xyxy Target rate is at least 70% and at most 85% of
maximum rate. The graph of the system is shown below.
b) From the graph you can see that the target zone for a 40-year-old person is between 126 and
153, inclusive. That is, 126≤y≤153. A 40-year-old person who has a heart rate of 150 is
within the target zone
Exercise
1) Test whether the ordered pair is a solution of the system 22
1
xy
x (-1, 2), (0, 0), (1, 4) and (2, 7)
2) Test whether the ordered pair is a solution of the corresponding system of inequality graphed
below? (25, -5), (2, 3) and (2, 6)
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3) Graph the system of linear inequalities a) 22
1
xy
x b)
1
3
y
yx c)
5
0
xy
x
4) Give an ordered pair that is a solution of the system;
a) 15
3
y
yx b)
2
6
x
yx c)
12
4
x
yx d)
yx
y
x
10
7
e)
132
5
3
yx
y
x
f)
xy
y
x
0
0
5) To be a flight attendant, you must be at least 18 years old and at most 55 years old, and you must
be between 60 and 74 inches tall, inclusive. Let X represent a person’s age (in years) and let y
represent a person’s height (in inches). Write and graph a system of linear inequalities showing
the possible ages and heights for flight attendants.
6) Match the system of linear inequalities with its graph
a) 4
2
y
x b)
4
2
y
x c)
xy
y
x
0
3
d)
xy
y
x
3
0
e)
xy
y
x
1
1
3
f)
xy
y
x
1
1
1
7) Write a system of linear inequalities for the region
8) Graph the system of linear inequalities
a) xyxy ,3,4
b) 52,6,1 xyxy c) 33,335,632 yxyxyx
d) 13,1,04 yxyxyx
e) 2,5,12 xyxyx
f) 3,8,435 yyxyx
g) 10,1,2 xyxyx
h) xyyxy ,24,0
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i) xyyxyx 2,165,0
j) xyyxxy ,15,9,0
k) 22,41 yxyx
l) 3,12,56 xyyxy
9) You are a lifeguard at a community pool, and you are in charge of maintaining the proper pH
(amount of acidity) and chlorine levels. The water test-kit says that the pH level should be
between 7.4 and 7.6 pH units and the chlorine level should be between 1.0 and 1.5 PPM (parts per
million). Let p be the pH level and let c be the chlorine level (in PPM). Write and graph a system
of inequalities for the pH and chlorine levels the water should have.
10) For a healthy person who is 4 feet 10 inches tall, the recommended lower weight limit is about 91
pounds and increases by about 3.7 pounds for each additional inch of height. The recommended
upper weight limit is about 119 pounds and increases by about 4.9 pounds for each additional
inch of height. Source: Dietary Guidelines Advisory Committee
a) Let x be the number of inches by which a person’s height exceeds 4 feet 10 inches and let y
be the person’s weight in pounds. Write a system of inequalities describing the possible
values of X and y for a healthy person.
b) Use a graphing calculator to graph the system of inequalities from Exercise 52.
c) What is the recommended weight range for someone 6 feet tall?
11) A shoe store gives a discount of between 10 to 25% on all sales. Let x be the regular footwear
price and Y be the discount price
a) Write a system of inequalities for the regular footwear prices and possible sale prices.
b) Graph the system you have written in part a above. Use your graph to estimate the range of
possible sale prices for shoes that are regularly priced at $65
12) The men’s world weightlifting records for the 105-kg-and-over weight category are shown in the
table. The combined lift is the sum of the snatch lift and the clean and jerk lift. Let s be the weight
lifted in the snatch and let j be the weight lifted in the clean and jerk. Write and graph a system of
inequalities to describe the weights you could lift to break the records for both the snatch and
combined lifts, but not the clean and jerk lift.
13) Each day, an average adult moose can process about 32 kilograms of terrestrial vegetation (twigs
and leaves) and aquatic vegetation. From this food, it needs to obtain about 1.9 grams of sodium
and 11,000 Calories of energy. Aquatic vegetation has about 0.15 gram of sodium per kilogram
and about 193 Calories of energy per kilogram, while terrestrial vegetation has minimal sodium
and about four times more energy than aquatic vegetation. Write and graph a system of
inequalities describing the amounts t and a of terrestrial and aquatic vegetation, respectively, for
the daily diet of an average adult moose.
14) A potter has 70 pounds of clay and 40 hours to make soup bowls and dinner plates to sell at a
craft fair. A soup bowl uses 3 pounds of clay and a dinner plate uses 4 pounds of clay. It takes 3
hours to make a soup bowl and 1 hour to make a dinner plate. If the profit on a soup bowl is $25
and the profit on a dinner plate is $20, how many bowls and plates should the potter make in
order to maximize profit?
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SEQUENCES AND SERIES Introduction Saying that a collection of objects is listed “in sequence” means that the collection is ordered so
that it has a first member, a second member, a third member, and so on. Below are two examples
of sequences of numbers. The numbers in the sequences are called terms
Sequence 1: 3 6 9 12 15 Sequence 2: 3 6 9 12 15 . . . . .
A sequence can be defined as an arrangement of numbers in a particular/ specific order. It can
also be defined as a function whose domain is a set of consecutive integers.
DOMAIN: 1 2 3 4 5 The domain gives the relative position of each term: 1st, 2nd, 3rd, and
so on. If a domain is not specified, it is understood that the domain starts with 1.
RANGE: 3 6 9 12 15 The range gives the terms of the sequence.
Sequence 1 above is a finite sequence because it has a last term. Sequence 2 is an infinite
sequence because it continues without stopping.
Each number in a sequence is called a term. 21,TT and n nT are used to denote the 1st, 2nd and
nth terms respectively.
Example 1 Determine the next 2 terms in each of the following sequences; a)
... 14, 11, 8, 5, b) … 22,15,9,4, c) ,...,,,181
41
21 d) ,...,,,
54
43
32
21
Solution
a) The difference between the terms is 3 therefore the next 2 terms are 17 and 20.
b) The difference between the terms is increasing by 1 so the next 2 terms are 30 and 39.
c) Ratio between consecutive terms is 21 the next 2 terms are
161 and
321 .
d) Notice that the denominator exceeds the numerator by 1. Thus the next 2 terms are 65 and
76
Example 2 List the 1st 5 terms of the sequence whose nth term is; a) 32T nn b) 1)2(T n
n
Solution
a) 53)1(21 T 73)2(22 T 93)3(23 T 113)4(24 T 133)5(25 T
So the sequence is 5 7 9 11 13 . . . . .
b) 1)2( 11
1 T 2)2( 12
2 T 4)2( 13
3 T 8)2( 14
4 T 16)2( 15
5 T
Therefore the sequence is 1 2 4 8 16 . . . . .
Remarks If the terms of a sequence have a recognizable pattern, then you may be able to write a
rule for the nth term of the sequence.
Example 3 For each sequence, describe the pattern, write the next term, and write a formula for
the nth term a) . . . 81
1 ,
27
1- ,
9
1 ,
3
1 b) 2 6 12 20 . . .
Solution
a) We can write the terms as . . . )( )( )( )( 4
313
312
311
31 The next term is
24315
31
5 )(T
. A formula for the nth term is n
n )(T31
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b) We can write the terms as 1(2), 2(3), 3(4), 4(5), . . . . The next term is 30)6(5T5 . A rule
for the nth term is )1(T nnn .
When the terms of a sequence are added, the resulting expression is a series. A series can be
finite or infinite. Eg Finite series 1512963 infinite series .....1512963
We can use summation notation to write a series. For example, for the finite series shown above,
we can write
5
1
3i1512963i
where i is the index of summation,
The summation notation is read as “the sum from i equals 1 to 5 of 3i.” Summation notation is
also called sigma notation because it uses the uppercase Greek letter sigma, written .
Summation notation for an infinite series is similar to that for a finite series. For example, for the
infinite series shown above, we can write
1
3i.....1512963i
. The infinity
symbol, , indicates that the series continues without end.
Example 4 Write each series with summation notation
a) 100....15105 b) ....54
43
32
21
Solution
a) Notice that the first term is 5(1), the second is 5(2), the third is 5(3), and the last is 5(20). So,
the terms of the series can be written as ii 5T where i = 1, 2, 3, . . . . 20 and the summation
notation for the series is
20
1
5ii
b) Notice that for each term the denominator of the fraction is 1 more than the numerator. So,
the terms of the series can be written as 1
T
i
ii where i = 1, 2, 3, 4, . . and the summation
notation for the series is
11i
i
i
Note The index of summation does not have to be i — any letter can be used. Also, the index
does not have to begin at 1. For instance, in part (b) of Example 5 on the next page, the index
begins at 3.
Example 5 Find the sum of the series a)
6
1
2i
i b)
6
3
2 )2(k
k
a) 4212108642)6(2)5(2)4(2)3(2)2(2)1(226
1
i
i
b) 9438271811)62()52()42()32()2( 22226
3
2 k
k
The sum of the terms of a finite sequence can be found by simply adding the terms. For
sequences with many terms, however, adding the terms can be tedious. Formulas for finding the
sum of the terms of four special types of sequences are given below.
nn
i
1
1 )1(21
1
nnin
i
)12)(1(61
1
2
nnnin
i
22122
41
1
3 )1()1(
nnnnin
i
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In words, the 1st formula gives the sum of n 1’s. The 2nd gives the sum of the positive integers
from 1 to n. The 3rd formula gives the sum of the squares of the positive integers from 1 to n.
Exercise
1) Write the first six terms of the sequence a) 13T nn b) nn 2T c) 32T n
n d)
1T 2 nn e) n
nn
2
2T
f) 2)1(T nn g) 1
32)(3T n
n
2) Write the next term in the sequence. Then write a rule for the nth term.
a) .....7531
b) ....1000100101
c) ....1410842
d) .....2015105-
e) ...81
61
41
21
f) .....85
74
63
52
41
g) .....35
34
33
32
31
h) .....
504
403
302
201
i) ....1.53.45.37.29.1
3) Write the series with summation notation.
a) 1713951
b) 20161284
c) ....211512933-
d) ....-54-32-1
e) 11-10-9-8-7-
f) ....98
87
76
65
g) 001.001.01.01
h) 362516941
4) Find the sum of the series a)
7
1
3i
i b)
3
1
34n
n c)
5
1
2 )1(k
k d)
4
0
2 )12(n
n e)
10
2
2
nn
f)
4
1
)2(k
kk g)
12
21
1
nn
h)
5
11
nnn i)
6
2 1i i
i
5) The diagram shows part of a roof frame.
The length (in feet) of each vertical
support is given below the support.
These lengths form an arithmetic
sequence from each end to the middle.
a) Find the total length of the vertical
supports from one end to the middle.
b) Use your result from Exercise 65 to
find the total length of the vertical
supports from end to end.
6) Use one of the formulas for special series to find the sum: a)
42
1
1i
b)
10
1n
n c)
12
1
2
i
i d)
6
1
3
k
k
7) Suppose you are stacking tennis balls in a pyramid as a display at a sports store. If the base is
an equilateral triangle, then the number na of balls per layer would be )1(21 nnan where n
= 1 represents the top layer. How many balls are in the fifth layer? How many balls are in a
stack with 5 layers?
Arithmetic Progression (AP)
In an arithmetic progression the difference between consecutive terms is constant. The constant
difference is called the common difference and is usually denoted by d.
The nth term and the sum of the 1st n terms of an AP with first term a and common difference d,
are respectively given by
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dnaTn )1( and dnan
sn )1(22
Example 1 Decide whether each sequence is arithmetic.
a) -3, 1, 5, 9, 13, . . . b) 2, 5, 10, 17, 26, . . .
Solution
To decide whether a sequence is arithmetic, find the differences of consecutive terms.
a) ...4 45342312 TTTTTTTT Each difference is 4, so the sequence is arithmetic
b) 9753 5342312 TTTTTTTT The differences are not constant, so the
sequence is not arithmetic.
Example 2 Find the 10th term and the sum of the first 20 terms of the AP ...,5,2,1,4
Solution
4903)120()42[(2
20 and 233)110(434 2010 sTda
Example 3 A sequence is given by the formula 53 nTn for ...3,,2,1n Write down the
first 4 terms of this sequence hence find S12 and T15
Solution
50)3(148 and 294)]3(11)8(2[2
1231714118 15124321 TSdTTTT
Example 4 The 10th term of an AP is -15 and the 31st term is -57 find the 15th term.
Solution
25)2(143 thus242215730
15915
31
10
Tdd
daT
daT
Example 5 Which term of the AP ...17,11,5 is 119?
Solution
201206119)1(5565 nnnTda n
Thus 119 is the 20th term of the given arithmetic progression ...17,11,5 .
Example 6 The sum of the first n terms of a certain progression is given by nnSn 72 . What
kind of a sequence is the progression?
Solution
246219
46144671
333213
22212
2
11
TTTTTS
TTTTSTS
The sequence is ...,2,4,6 which is an AP with a common difference .2d
Example 7 The first row of a concert hall has 25 seats, and each row after the first has one
more seat than the row before it. There are 32 rows of seats.
a) Write a rule for the number of seats in the nth row.
b) 35 students from a class want to sit in the same row. How close to the front can they sit?
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c) What is the total number of seats in the concert hall?
d) Suppose 12 more rows of seats are built (where each row has one more seat than the row
before it). How many additional seats will the concert hall have?
Solution
a) Use 25a and 1d to write a formula for the nth term
nndna 241)1(25)1(Tn
b) Using n 24Tn put 35Tn nad solve for n. ie 112435 nn . The class can sit in
the 11th row
c) Find the sum of an arithmetic series with 25a 1d and 32n . Ie
1296)3150(16]1)132()25(2[S])1(2[S232
322n dnan
There are 1296 seats in the concert hall.
d) The expanded concert hall has 441232 n rows of seats. Because 25a and 1d ,
the total number of seats in the expanded hall is
2046)4350(22]1)144()25(2[S244
44
The number of additional seats is 7501296-2046SS 3244 .
Exercise
1) Decide whether the sequence is arithmetic. Explain why or why not.
a) ....2581114
b) ....8127931
c) ...15-13-11-7-5-
d) ....2.521.510.5
e) ....5
1658
54
52
51
f) ....1131
31
35
2) Which term of the sequence ....,,0,,3,-23
23
29 is 27?
3) Find the sum of the first 10 terms of the arithmetic series;
a) ....18141062
b) ....54.543.53
c) ....(-6)(-3)036
d) ....5.54.33.11.90.7
4) The nth term of a sequence is given by the formula below. Write down the first 4 terms of this
sequence hence find S12 and T15
a) nTn 27
b) 35 nTn
c) nTn 25
d) nTn 312
e) nTn 214
f) 25.045.0 nTn
5) Suppose a movie theater has 42 rows of seats and there are 29 seats in the first row. Each row
after the first has two more seats than the row before it. How many seats are in the theater?
6) Write a rule for the nth term of the arithmetic sequence. Then find the 25th term
a) ....97531
b) ....3022146
c) ....5137239
d) ....32101
e) ...83221
211
21
f) ....5214
g) ....23
617
625
211
h) ....21
67
611
25
i) ....8.84.646.1
7) Write a formula for the nth term of the arithmetic sequence
a) 46T and 4 14 d
b) 80 and 12 ad
c) 24T and 835 d
d) 77T and 17T 155
e) 4T and 6 12 d
f) 52T and 28T 202
g) 61
9T and 2 a
h) 122T and 34T 187
i) 2.48T and 1.4 16 d
8) For part (i), find the sum of the first n terms of the arithmetic series. For part (ii), find n for
the given sum nS
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 22
a) 366S ii) 20 i) ....23181383 nn
b) 182S ii) 40 i) .. ..1826344250 nn
c) 375S ii) 19 i) ....1050)5(10- nn
d) -12S ii) 23 i) ....2225283134 nn
e) 6611S ii) 68 i) ....30231692 nn
f) 2178S ii) 24 i) ....584430162 nn
9) Which term of the sequence ....75.542.5 is 31? Find also the 12th term and the
sum of the first 15 terms.
10) The sum of the first n terms of a certain progression is given by; a) nnSn 23 b)
28 nnSn c) nnSn 72 what kind of a progression is this?
11) Three numbers are in an AP. The difference between the 1st and 3rd number is 8. If the product of
these 2 numbers is 20, find the 3 numbers.
12) The distance covered by a rolling object at intervals of 1 second was recorded as 4cm, 16cm,
28cm, 40cm and so on. How long will the object be 22.6m from the starting point?
13) Evaluate; a)
20
1
5i)(3i
b)
15
1
3i)-(-10i
c)
22
1
43 i)-(6
i
d)
43
1
i)4(11i
e)
18
1
i)4.4(8.1i
14) Domestic bees make their honeycomb
by starting with a single hexagonal cell,
then forming ring after ring of hexagonal
cells around the initial cell, as shown.
The numbers of cells in successive rings
form an arithmetic sequence.
a) Write a rule for the number of cells in the nth ring.
b) What is the total number of cells in the honeycomb after the 9th ring is formed? (Hint:
Do not forget to count the initial cell.)
15) Logs are stacked in a pile, as shown at
the right. The bottom row has 21 logs
and the top row has 15 logs. Each row
has one less log than the row below it.
How many logs are in the pile? 2
16) Suppose each seat in rows 1 through 11 of the concert hall in Example 7 costs $24, each seat
in rows 12 through 22 costs $18, and each seat in rows 23 through 32 costs $12. How much
money does the concert hall take in for a sold-out event?
17) A quilt is made up of strips of cloth,
starting with an inner square surrounded
by rectangles to form successively larger
squares. The inner square and all
rectangles have a width of 1 foot. Write
an expression using summation notation
that gives the sum of the areas of all the
strips of cloth used to make the quilt
shown. Then evaluate the ex pression
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 23
18) A paper manufacturer sells paper rolled onto cardboard dowels. The thickness of the paper is
0.004 inch. The diameter of a dowel is 3 inches, and the total diameter of a roll is 7 inches as
shown.
a) Let n be the number of times the
paper is wrapped around the dowel,
let Tn be the diameter of the roll just
before the nth wrap, and let ln be the
length of paper added in the nth
wrap. Copy and complete the table
a) What can you say about the sequence ....,,,, 4321 llll ? Write a formula for the nth
term of the sequence.
b) Find the number of times the paper must be wrapped around the dowel to create a roll
with a 7 inch diameter. Use your answer and the formula from part (b) to find the length
of paper in a roll with a 7 inch diameter.
c) Suppose a roll with a 7 inch diameter costs $15. How much would you expect to pay for
a roll with an 11 inch diameter whose dowel also has a diameter of 3 inches? Explain
your reasoning and any assumptions you make.
19) One of the major sources of our knowledge of Egyptian mathematics is the Ahmes papyrus
(also known as the Rhind papyrus), which is a scroll copied in 1650 B.C. by an Egyptian
scribe. The following problem is from the Ahmes papyrus. Divide 10 hekats of barley among
10 men so that the common difference is81 of a hekat of barley. Use what you know about
arithmetic sequences and series to solve the problem.
Geometric Progression (GP)
It is a sequence in which the ratio between any two consecutive terms is a constant. This ratio is
called the common ratio and is denoted by r. Thus ...3
4
2
3
1
2
term
term
term
term
term
termr
rd
th
nd
rd
st
nd
Examples of G.P are a) ...,8,4,2,1 b) ...,,1,391
31 c) ...,27,9,3,1 d) ...,,,,, 432 xxxx
Generally a G.P will be of the form ...,,,, 32 ararara . The nth term and the sum of the first n
terms of a geometric sequence with first term a and common ratio r are respectively given by;
1 n
n arT and r
raS
n
n
1
)1(
However if 0a and 1|| r then r
aS
1 called the sum to infinity/limiting sum of the G.P.
Example 1 Decide whether each sequence is geometric.
a) 1 , 2 , 6 , 24 , 120, . . . . b) 81 , 27 , 9 , 3 , 1, . . .
Solution
To decide whether a sequence is geometric, find the ratios of consecutive terms.
a) 54
5...4
3
43
2
32
1
2
term
term
term
term
term
term
term
termth
th
rd
th
nd
rd
st
nd
The ratios are different, so the sequence is not geometric.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 24
b) 3
1
4
5...
3
1
9
3
3
4
3
1
27
9
2
3
3
1
81
27
1
2
term
term
term
term
term
term
term
termth
th
rd
th
nd
rd
st
nd
The ratios are the same, so the sequence is geometric
Example 2 Find the 10th term and the sum of the first 10 terms of the G.P ...,,,181
41
21 Also
find the sum to infinity.
Solution
3
2
)(1
1 and
512
341
)(1
])(1[1,)(1 and 1
21
21
10
21
1051219
21
1021
SSTra
Example 3 The 4th and 9th terms of a G.P are 8 and 256 respectively. Determine the 6th term and
the sum of the first 7 terms.
Solution
1 so 2328
256T and 8T 5
3
88
9
3
4 arrar
ararar
12721
)21(1S and 32)2(1T
7
7
5
6
Example 4 For the geometric sequence . . . 40- 20 10- 5 find the terms whose value is 320.
Solution
761)2(64)2(320)2(5T2 and 5 611
n nnra nn
Example 5 The ratio between the 7th and 5th terms of a geometric sequence is 4. If the 3rd term
is 8, find the 15th term and the sum of the first 8 terms.
Solution
14
15
2
3
2
4
6
5
7 )2(2T284Tbut 24T
Taaarrr
ar
ar
17021
])2(1[2S ,2 when and 510
21
])2(1[2S ,2 when Now
8
8
8
8
rr
Example 6 In 1990 the average monthly bill for cellular telephone service in the United States
was $80.90. From 1990 to 1997, the average monthly bill decreased by about 8.6% per year.
a) Write a rule for the average monthly cellular telephone bill an (in dollars) in terms of the
year. Let n = 1 represent 1990.
b) What was the average monthly cellular telephone bill in 1993?
c) When did the average monthly cellular telephone bill fall to $50?
d) On average, what did a person pay for cellular telephone service during 1990–1997?
Solution
a) Because the average monthly bill decreased by the same percent each year, the average
monthly bills from year to year form a geometric sequence. Use a = 80.9 and
914.0086.01 r , a rule for the average monthly bill is 1)914.0(9.80T n
n
b) In 1993, n = 4. So, the average monthly bill was 77.61$)914.0(9.80T 3 n .
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 25
c) We want to find n such that 50T n . Therefore
65914.0ln
618.0ln10.618
9.80
50)914.0()914.0(9.8050 11 nnnn
The average monthly cellular telephone bill reached $50 in 1995 (when n = 6).
d) Because the model 1)914.0(9.80T n
n gives the average monthly bill, the model
11 )914.0(8.970)914.0(9.8012Y nn
n gives the average annual bill. Using a = 970.8 and
914.0r , we can estimate a person’s total cost for cellular telephone service during the 8
year period 1990–1997 to be
5790914.01
)914.0(18.970S
8
n
A person paid about $5790 for cellular telephone service during 1990–1997.
Exercise
1) Decide whether the sequence is arithmetic, geometric, or neither. Explain your answer
a) .. . . 384 96 24 6 d) . . . . 9- 5- 1- 3 g) ....134
32
31
b) . . . . 13 7 3 1 e) . . . 1 3- 7- 11- h) ...323
161
81
43
c) . . . . 31 22 13 4 f) ....227
29
23
21 i) ...
6256
1255
254
53
2) te a rule for the nth term of the geometric sequence below then find 6T .
a) . . . . 64- 16 4- 1 c) . . . . 686 98 14 2 e) . . . . - - 5275
95
35
b) . . . . 40 20 10 5 d) . . . . 750- 150 50- 6 f) . . . . 22716
98
34
3) Write a rule for the nth term of the geometric sequence.
a) 4 and 3 ar
b) 45 and 31 ar
c) 72T and 6 3 r
d) 4 and 81 ar
e) 2 and 8 ar
f) 16T and 421 a
g) 300T & 10T 63
h) 5T & 20T 42
i) 3750T& 30T 52
4) For part (i), find the sum of the first n terms of the geometric series. For part (ii), find n for
the given sum Sn
a) 341S ii) 14 i) . . . . 641641 nn
b) 208S ii) 10 i) . . . . 7291891 nn
c) 3829S ii) 18 i) . . . . (-189)63)21(7 nn
d) 67.66S ii) 16 i) . . . . 10)(3090-3
10 nn
5) Find the sum of the series; a)
10
1
16(2)i
i b)
8
1
15(4)i
i c)
9
121 )12(-
i
i d)
10
1
18(0.75)i
i
e)
6
123 )4(
i
i f)
12
1
1(-2)i
i
6) The ratio between the 5th and 2nd terms of a GP is 827 if the 3rd term is 4.5 find the 10th term
and the sum of the first 10 terms.
7) The men’s U.S. Open tennis tournament is held annually in Flushing Meadow in New York
City. In the first round of the tournament, 64 matches are played. In each successive round,
the number of matches played decreases by one half.
a) Find a rule for the number of matches played in the nth round.
b) For what values of n does your rule make sense?
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 26
c) Find the total number of matches played in the men’s U.S. Open tennis tournament.
8) Which term of the GP . . . . 313
131 is 81? Also find the sum upto this term and the
limiting sum of the GP.
9) The sum to infinity of a certain GP is 8 and the sum of the 1st 2 terms is38 , find the 1st term and
the sum of the fist 4 terms.
10) When a computer must find an item in an ordered list of data (such as an alphabetical list of
names), it may be programmed to perform a binary search. This search technique involves
jumping to the middle of the list and deciding whether the item is there. If not, the computer
decides whether the item comes before or after the middle. Half of the list is then ignored on
the next pass through the list, and the computer jumps to the middle of the remaining list.
This is repeated until the item is found.
a) An ordered list contains 1024 items. Find a rule for the number of items remaining after
the nth pass through the list.
b) In the worst case, the item to be found is the only one left in the list after n passes through
the list. What is the worst-case value of n for a binary search of a list with 1024 items?
11) In 1990 factory sales of pagers in the United States totaled $118 million. From 1990 through
1996, the sales increased by about 20% per year.
a) Write a rule for pager sales Tn (in $”000,000) in terms of the year. Let n = 1 rep 1990.
b) What did factory sales of pagers total in 1992?
c) When did factory sales of pagers reach $300 million?
d) What was the total of factory sales of pagers for the period 1990–1996?
12) The Sierpinski triangle is a design using equilateral triangles. The process involves removing
smaller triangles from larger triangles by joining the midpoints of the sides of the larger
triangles as shown below. Assume that the initial triangle is equilateral with sides 1 unit long.
a) Let Tn be the number of triangles removed at the nth stage. Find a rule for Tn. Then find
the total number of triangles removed through the 10th stage.
b) Let An be the remaining area of the original triangle at the nth stage. Find a rule for An.
Then find the remaining area of the original triangle at the 15th stage.
13) Suppose two computer companies, Company A and Company B, opened in 1991. The
revenues of Company A increased arithmetically through 2000, while the revenues of
Company B increased geometrically through 2000. In 1996 the revenue of Company A was
$523.7 million. In 1996 the revenue of Company B was $65.6 million.
a) The revenues of Company A have a common difference of 55.5. The revenues of
Company B have a common ratio of 2. Find a rule for the revenues in the nth year of each
company. Let a represent 1991.
b) Find the sum of the revenues from 1991 through 2000 for each company.
c) Find the year when the revenue of Company B exceeds the revenue of Company A. Write
a brief paragraph explaining which company you would rather own.
14) Using the rule for the sum of the first n terms of a geometric series, write the polynomial as a
rational expression a) 4321 xxxx b) 753 241263 xxxx
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 27
MATRICES A matrix (in plural matrices) is a rectangular arrangement of numbers in rows and columns. For
instance, matrix A below has two rows and three columns.
The dimension of this matrix are 32 (read
“2 by 3”). The numbers in a matrix are its
entries. In matrix A, the entry in the second
row and third column is 5.
rows 2502
126
column 3
A
Special Matrices Some matrices have special names because of their dimensions or entries
i) Row matrix: a matrix with only one row. Example 4023
ii) Column matrix: a matrix with only one column. Example
1
2
iii) Square matrix: a matrix whose number
of rows equals the number of columns.
Example
233
163
121
iv) Zero matrix: a matrix whose entries are
all zeros. Example
00
00
Equality of matrices
Two matrices are equal if their dimensions are the same and the entries in corresponding
positions are equal.
Example Given
42
33
42
83
x
x
y
x, find the values of x and y
Solution The two matrices are equal corresponding entries are equal. So
3622 and 48238 yxyxxxx
Addition and Subtraction of Matrices To add or subtract matrices, you simply add or subtract corresponding entries.
You can add or subtract matrices only if they have the same dimensions
Example Perform the indicated operation, if possible.
a)
3
0
1
7
4
3
b)
16
72
04
38 c)
5
1
43
02
S0lution
a) Since the matrices have the same dimensions, you can add them.
5
1
4
32
54
13
3
5
1
2
4
3
b) Since the matrices have the same dimensions, you can subtract them.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 28
12
66
)1(064
)3(328
16
32
04
38
c) Since the order of
43
02is 22 and the order of
5
1is 12 ,we cannot add the matrices.
Remark Addition of matrices is both commutative and associative that is, if A, B, and C are
matrices with the same dimensions When adding matrices, you can regroup them and change
their order without affecting the result.
A + B = B + A commutative property
(A + B) + C = A + (B + C) Associative property
Scalar multiplication
In matrix algebra, a real number is often called a scalar. To multiply a matrix by
a scalar, you multiply each entry in the matrix by the scalar. This process is called scalar
multiplication
Note Multiplication of a sum or difference of matrices by a scalar obeys the distributive
property. That is B +A )B +A ( kkk and B -A )B -A ( kkk
Example 1 Perform the indicated operation(s) a)
24
02
2
3 b)
62
86
54
54
30
21
2
S0lution
a)
36
03
24
02
24
02
2
3
2
3
2
3
2
3
2
3
b)
46
146
96
62
86
54
108
60
42
62
86
54
54
30
21
2
You can use what you know about matrix operations and matrix equality to solve a matrix
equation.
Example 2 Solve the matrix equation for x and y:
812
026
2
14
58
132
y
x
S0lution
812
026
21012
086
56
0432
2
14
58
132
y
x
y
x
y
x
Equate corresponding entries and solve the two resulting equations.
1 and38210 and2686 yxyx
Example 3 Use matrices to organize the following information about health care plans.
This Year For individuals, Comprehensive, HMO Standard, and HMO Plus cost
$694.32, $451.80, and $489.48, respectively. For families, the Comprehensive, HMO
Standard, and HMO Plus plans cost $1725.36, $1187.76, and $1248.12.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 29
Next Year For individuals, Comprehensive, HMO Standard, and HMO Plus will cost
$683.91, $463.10, and $499.27, respectively. For families, the Comprehensive, HMO
Standard, and HMO Plus plans will cost $1699.48, $1217.45, and $1273.08.
S0lution
One way to organize the data is to use 23 matrices, as shown.
This Year (A) Next Year (B)
Plus HNO
Standard HNO
iveComprehens
$1248.12 48.489$
$1187.76 80.451$
$1725.36 32.694$
Family Individual
$1273.08 27.499$
$1217.45 10.463$
$1699.48 91.683$
Family Individual
Note We can also organize the data using 32 matrices where the row labels are levels of
coverage (individual and family) and the column labels are the types of plans(Comprehensive,
HMO Standard, and HMO Plus).
Example 4 A company offers the health care plans in Example 3 above to its employees. The
employees receive monthly paychecks from which health care payments are deducted. Use the
matrices in Example 3 to write a matrix that shows the monthly changes in health care payments
from this year to next year.
S0lution
Begin by subtracting matrix A from matrix B to determine the yearly changes in health care
payments. Then multiply the result by 12
1 and round answers to the nearest cent to find the
monthly changes
$2.08 82.0$
$2.47 94.0$
$2.16- 87.0$
$24.96 79.9$
$29.69 30.11$
$25.88- 41.10$
12
1
$1248.12 48.489$
$1187.76 80.451$
$1725.36 32.694$
$1273.08 27.499$
$1217.45 10.463$
$1699.48 91.683$
12
1)A - B(
12
1
The monthly deductions for the Comprehensive plan will decrease, but the monthly deductions
for the other two plans will increase.
Exercise
1) Perform the indicated operation(s), if possible
6-
10-
11-
9
22-
20
a-) b)
9-129
85-1-
1-04-
47-6- c)
54-
024- d)
3-4
01-5
02
1-5-6
2) In Example 3, suppose the annual health care costs given in matrix B increase by 4% the
following year. Write a matrix that shows the new monthly payment.
3) Perform the indicated operation, if possible. If not possible, state the reason.
11
54
66
28 a) b)
94
72
10
53 c)
3.56.2
7.81.4
1.52.0
5,32.1
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 30
5110
363
2911
417 d) e)
5
2
832
1
4
3
4
1
2
1
f)
2111
972
937
82
61
61
4) Perform the indicated operation(s), if possible
604-
73-1- 4 -a) b)
1-3
6-1-5 c)
11-5-3-
1555-
931
4 d)
94
31
2-1-
00
9- e)
10-
42-2-
2
1
116
41
8-10
4.48.4-
1.5-2.1
4.36.8-
5.2 f) g)
54
23
01
4
30
50
812
h)
1963
1351
474
21062
65
424
31
772 i) j)
553
7142
268
0173
5) Solve the matrix equation for x and y.
a)
910
6
910
82 yx
b)
0
216
87
04
81
23
y
x
c)
20
1612
511
432
yx
d)
75
0
81
94
77
5
21
78
34
y
x
6) Use the following information about three Major League Baseball teams’ wins and losses in
1998 before and after the All-Star Game. Before The Atlanta Braves had 59 wins and 29
losses, the Seattle Mariners had 37 wins and 51 losses, and the Chicago Cubs had 48 wins
and 39 losses. After The Atlanta Braves had 47 wins and 27 losses, the Seattle Mariners had
39 wins and 34 losses, and the Chicago Cubs had 42 wins and 34 losses.
a) Use matrices to organize the information.
b) Using your matrices from Exercise 37, write a matrix that shows the total numbers of
wins and losses for the three teams in 1998.
7) The matrices below show the average yearly cost (in dollars) of tuition and room and board
at colleges in the United States from 1995 through 1997. Use matrix addition to write a
matrix showing the totals of these costs. _Source: U.S. Department of Education TUITION ROOM AND BOARD
Collegeyear - 4 Private
Collegeyear - 2 Private
Collegeyear - 4 Public
Collegeyear - 2 Public
920,12243,12481,11
190,7094,7914,6
986,2848,2681,2
283,1239,1192,1
199719961995
555,5368,5121,5
699,4469,4256,4
345,4166,4990,3
128,3978,2944,2
199719961995
8) The figures below give the number (in millions) of Hispanic CD, cassette, and music video
units shipped to all market channels and the dollar value (in millions) of those shipments (at
suggested list prices).
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 31
1996 Number of units—CDs:
20,779; cassettes: 15,299; and music
videos: 45.
Dollar value—CDs: 268,441;
cassettes: 122,329; and music
videos: 916.
1997 Number of units—CDs:
26,277; cassettes: 17,799; and music
videos: 70.
Dollar value—CDs: 344,697;
cassettes: 144,645; and music
videos: 1,260.
a) Use matrices to organize the
information.
b) Write a matrix that gives the total
numbers of units shipped and total
values for both years.
c) Write a matrix that gives the change
in units shipped and dollar value
from 1996 to 1997.
9) Eligibility for a National Merit Scholarship is based on a student’s PSAT score. Through
1996, this total score was found by doubling a student’s verbal score and adding this value to
the student’s mathematics score. Let V represent the average verbal scores and let M
represent the average mathematics scores earned by sophomores and juniors at Central High
for tests taken in 1993 through 1996.
VERBAL SCORES (V) MATHEMATIS SCORES (M)
1996
1995
1994
1993
6.482.48
8.487.48
9.489.48
0.499.48
Juniors Sophonore
9.508.49
8.504.49
0.503.48
4.500.49
Juniors Sophonore
a) Write an expression in terms of V and M that you could use to determine the average total
PSAT scores for sophomores and juniors at Central High from 1993 through 1996. Then
evaluate the expression.
b) Use the matrix from Exercise 43 to determine the average total PSAT score for juniors at
Central High in 1996.
10) A triangle has vertices (2, 2), (8, 2), and (5, 6). Assign a letter to each vertex and organize the
triangle’s vertices in a matrix. When you multiply the matrix by 4, what does the “new”
triangle look like? How are the two triangles related? Use a graph to help you.
11) The matrices show the number of people (in thousands) who lived in each region of the
United States in 1991 and the number of people (in thousands) projected to live in each
region in 2010. The regional populations are separated into three age categories
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 32
1991 2010
Pacific
Moutains
South
Midwest
Northwest
331.4001.25603.10
580.1461,8903.3
942.10474.53504.22
857.7554.36814.15
043.7391.31142.12
65Over 6518170
551.5125.31655.13
707.2420,1294.5
832.14557.67128.25
980.8095.41840.15
377.7822.38493.12
65Over 6518170
a) The total population in 1991 was 252,177,000 and the projected total population in 2010
is 297,716,000. Rewrite the matrices to give the information as percents of the total
population. (Hint: Multiply each matrix by the reciprocal of the total population (in
thousands), and then multiply by 100.)
b) Write a matrix that gives the projected change in the percent of the population in each
region and age group from 1991 to 2010.
c) Based on the result of Exercise 46, which region(s) and age group(s) are projected to
show relative growth from 1991 to 2010?
12) The matrices show the number of hardcover volumes sold and the average price per volume
(in dollars) for different subject areas.
_ 1995 (A) 1996 (B)
per volume sold
pric Average volume
per volume sold
pric Average volume
Travel
M usic
Law
Art
30.38000,199
27.43000,251
09.73000,716
23.41000,116,1
92.33000,179
21.39000,253
51.88000,827
40.53000,070,1
b) Calculate B o A. How many more (or fewer) law volumes were sold in 1996 than in
1995? How much more (or less) did the average music book cost in 1996 than in 1995?
c) Calculate B + A. Does the “volumes sold” column in B + A give you meaningful
information? Does the “average price per volume” column in B + A give you meaningful
information? Explain.
d) What conclusions can you make about the number of volumes sold and the average price
per volume of these books from 1995 to 1996?
Multiplying Matrices The product of two matrices A and B is
defined provided the number of columns in
A is equal to the number of rows in B. If A
is an nm matrix and B is an pn matrix,
then the product AB is an pm matrix.
Example 1 State whether the product AB is defined. If so, give the dimensions of AB.
a) A: 32 , B: 43 b) A: 23 , B: 43
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Solution
a) Because A is a 32 matrix and B is a 43 matrix, the product AB is defined and is a 42
matrix.
b) Because the number of columns in A (two) is not equal to the number of rows in B (three),
the product AB is not defined.
Example 2 Given matrices
06
4-1
32-
A and
42-
31-B , find the product AB if possible
Solution
Because A is a 23 matrix and B is a 22 matrix, the product AB is defined and is a 23
matrix. To write the entry in the first row and first column of AB, multiply corresponding entries
in the first row of A and the first column of B. Then add. Use a similar procedure to write the
other entries of the product.
186
13-7
64-
(4)03))(6((-2)0(6)(-1)
(-4)(4)3))(1((-2))4(1(-1)
3(4)3))(2(3(-2)(-2)(-1)
42-
31-
06
4-1
32-
AB
Example 3 If
01
23A and
12
41B , find the product AB and BA if possible
Solution
41
107
12
41
01
23AB
45
27
01
23
12
41BA
Notice that AB ≠ BA. Matrix multiplication is not, in general, commutative.
Example 4 Given
31
12A ,
24
02-B and
23
11C , obtain A(B+C) and AB+AC
Solution
1122
65
47
11-
31
12
23
11
24
02-
31
12C)A(B
1122
65
58
45
614
20
23
11
31
12
24
02-
31
12CAB A
Notice that A(B + C) = AB + AC, which is true in general. This and other properties of matrix
multiplication are summarized below
Remark Matrix multiplication is both associative and distributive. That is if A, B and C are
matrices then;
A(BC)=(AB)C :- associative property of matrix multiplication
A(B+C)=AB+AC :- left distributive property
(A+B)C=AC+BC :- right distributive property
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Matrix multiplication is useful in business applications because an inventory matrix, when
multiplied by a cost per item matrix, results in a total cost matrix.
pmpnnm
matrix
Cost Total
matrix
unitper Cost
matrix
Inventory
For the total cost matrix to be meaningful, the column labels for the inventory matrix must match
the row labels for the cost per item matrix.
Example 5 Two softball teams submit equipment lists for the season.
Women’s team 12 bates 45 balls 15 uniforms
Men’s team 15 bates 35 balls 17 uniforms
Each bat costs $21, each ball costs $4, and each uniform costs $30. Use matrix multiplication to
find the total cost of equipment for each team
Solution
To begin, write the equipment lists and the costs per item in matrix form. Because you want to
use matrix multiplication to find the total cost, set up the matrices so that the columns of the
equipment matrix match the rows of the cost matrix.
Equipment cost
teamsMen;
teamsWomen;
17 35 15
15 45 12
uniforms Balls Bates
Uniform
Ball
Bat
30$
4$
21$
The total cost of equipment for each team can now be obtained by multiplying the equipment
matrix by the cost per item matrix. The equipment matrix is 32 and the cost per item matrix is
13 , so their product is a 32 matrix.
977
882
)30(17 )4(35 )21(15
)30(15 )4(45 )21(12
30
4
21
17 35 15
15 45 12
The labels for the product matrix are as follows
teamsMen;
teamsWomen;
977$
882$
Cost Total
The total cost of equipment for the women’s
team is $882, and the total cost of equipment
for the men’s team is $977.
Exercise
1) State whether the product AB is defined. If so, give the dimensions of AB.
a) A: 23 , B: 32
b) A: 33 , B: 33
c) A: 23 , B: 23
d) A: 21 , B: 23
e) A: 42 , B: 34
f) A: 24 , B: 53
g) A: 55 , B: 45
h) A: 43 , B: 14
i) A: 33 , B: 42
2) Find the product.; a)
3
2
5- 3
4- 4 b)
3 1
0 2
1- 2-
0 1 c)
1- 2-
0 1
1- 0
2- 3
3 3-
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3) Use matrix multiplication to find the total cost of equipment in Example 5 if the women’s
team needs 16 bats, 42 balls, and 16 uniforms and the men’s team needs 14 bats, 43 balls,
and 15 uniforms.
4) Find the product. If it is not defined, state the reason.
a)
21-
0
12
- -3
1
2
1
6
1
b)
3.2 2.0
0 1.8
1.5 37.
-8.7- 2.6 4.2-
c)
3- 0
1- 4
2- 3
4- 1
d)
3 5-
4 1-
3 0
2- 6-
e)
5 2- 8
2- 1 0
2 5- 0
1 8- 2
f)
0.5- 1.5
0 1
0.3 2.9
0.2 0.2-
0 6.0
g)
0
0.2
1.2
0.25- 1.5- 1
1.25 0.5- 1-
h)
4 3 1-
4 2- 7-
3 1- 0
1 7 0.1
8 3 2-
1 1 6
i)
4 3 1-
4 2- 7-
3 1- 0
1 7 0.1
8 3 2-
1 1 6-
j)
1- 6- 4
5 2- 4-
5 0
4 1
2- 6
k)
12- 5- 4-
6 12 3
3 7- 5
3 5 2-
18- 3- 6
0 1 0
5) Using the given matrices, simplify the expression.
4 1 3
2 4 1-
6 5 2-
E
3 3- 2-
4 2 1-
1 2- 3
D1 2-
3 1-C
4 2-
0 1 B
1- 6
2- 4A a) AB
b) AB + AC c) D(D + E) d) (E + D)E e) -3AC f) 0.5AB+2AC
6) Solve for x and y; a)
y
x 19
6
3
1
4 2- 0
4 2 3
2 1 2-
b)
11 13-
5y
4 1-
1 2
2- 9
1 x 2-
3 1 4
7) The percents of the total 1997 world production of wheat, rice, and maize are shown in the
matrix for the four countries that grow the most grain: China, India, the Commonwealth of
Independent States (formerly the Soviet Union), and the United States. The total 1997 world
production (in thousands of metric tons) of wheat, rice, and maize is 608,846, 570,906, and
586,923 respectively.
a) Rewrite the matrix to give the
percents as decimals.
b) Show how matrix multiplication can
be used to determine how many
metric tons of all three grains were
produced in each of the four
countries.
GRAIN PRODUCTION
U.S
C.I.S
India
China
40.5 4.1 3.11
0.5 1.0 5.7
7.1 5.21 22
18 8.34 1.20
Maize Ricewheat
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8) In Exercises 37º39, use the following information. Three teams participated in a debating
competition. The final score for each team is based on how many students ranked first,
second, and third in a debate. The results of 12 debates are shown in matrix A.
MATRIX A
3 Team
12 Team
1 Team
2 6 4
5 2 5
4 5 3
3rd 2nd1st
a) Teams earn 6 points for each first place,
5 points for each second place, and 4
points for each third place. Organize this
information into a matrix B.
b) Find the product AB.
9) Which team won the competition? How
many points did the winning team score?
10) The numbers of calories burned by people of different weights doing different activities for
20 minutes are shown in the matrix.
Show how matrix multiplication can be
used to write the total number of calories
burned by a 120 pound person and a 150
pound person who each bicycled for 40
minutes, jogged for 10 minutes, and then
walked for 60 minutes.
CALORIES BURNED
person person
150lb 120lb
Walking
Jogging
Cycling
79 64
159 127
136 109
11) Matrix A is a 90° rotational matrix. Matrix B contains the coordinates of the triangle’s
vertices shown in the graph.
0 1
1- 0A
2 8 4
4- 4- 7B
a) Calculate AB. Graph the
coordinates of the vertices given
by AB. What rotation does AB
represent in the graph?
b) Find the 180° and 270° rotations
of the original triangle by using
repeated multiplication of the 90°
rotational matrix. What are the
coordinates of the vertices of the
rotated triangles?
Determinants and Cramer’s Rule Associated with each square matrix is a real number called its determinant The determinant of a
matrix A is denoted by det A or by |A|.
Determinant of a 22 Matrix
The determinant of a 22 matrix is the difference of the products of the entries on the diagonals
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Determinant of a 33 Matrix
Repeat the first two columns to the right of the determinant.
Subtract the sum of the products in red from the sum of the products in blue.
2
1 Example 1 Evaluate the determinant of the matrix. a)
52
31 b)
421
102
312
Solution
You can use a determinant to find the area of a triangle whose vertices are points in a coordinate
plane. The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by
1
1
1
2
1 Area
33
22
11
yx
yx
yx
where indicates
the absolute value for the determinant.
Example 2 The area of the triangle shown
is:
126
104
121
2
1 Area
Example 3 The Bermuda Triangle is a large triangular region in the Atlantic Ocean. Many
ships and airplanes have been lost in this region. The triangle is formed by imaginary lines
connecting Bermuda, Puerto Rico, and Miami, Florida. Use a determinant to estimate the area of
the Bermuda Triangle.
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Solution
The approximate coordinates of the Bermuda Triangle’s three vertices are (938, 454), (900,
o518), and (0, 0). So, the area of the region is as follows:
242,447)600,40800()00884,485(2
1
100
1518900
1454938
2
1 Area
The area of the Bermuda Triangle is about 447,000 square miles.
Cramer’s Rule
We can use determinants to solve a system of linear equations. The method, called Cramer’s
rule and named after the Swiss mathematician Gabriel Cramer (1704o1752), uses the coefficient
matrix of the linear system.
Linear system fdycx
ebyax
coefficient matrix
dc
ba
Cramer’s Rule for a 22 System
Let A be the coefficient matrix of this linear system: fdycx
ebyax
. If 0A , then the system has
exactly one solution. The solution is: df
bex
A
1 and
fc
eay
A
1
In Cramer’s rule, notice that the denominator for x and y is the determinant of the coefficient
matrix of the system. The numerators for x and y are the determinants of the matrices formed by
using the column of constants as replacements for the coefficients of x and y, respectively.
Example 4 Use Cramer’s rule to solve this system: 1042
258
yx
yx
Solution
Evaluate the determinant of the coefficient matrix
Coefficient matrix A is
42
58A and it’s determinant is 42
42
58A
Apply Cramer’s rule since the determinant is not 0.
1410
52
42
1
x and 2
102
28
42
1
y The solution is (o1, 2).
Check this solution in the original equations.
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Cramer’s Rule for a 33 System
Let A be the coefficient matrix of this linear system:
lizhygx
kfzeydx
jczbyax
If 0A , then the system has exactly one solution. The solution is:
ihl
fek
cbj
xA
1 ,
ilg
fkd
cja
yA
1 and
lhg
ked
jba
zA
1
Example 5 The atomic weights of three compounds are shown. Use a linear system and
Cramer’s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O)
Compound Formular Atomic weight
Methane 4CH 16
Glycerol 383 OHC 92
Water OH2 18
Solution
Write a linear system using the formula for each compound. Let C, H, and O represent the
atomic weights of carbon, hydrogen, and oxygen.
18O2H
92O3H83C
164HC
Evaluate the determinant of the coefficient matrix.
10
120
383
041
Apply Cramer’s rule since the determinant is not 0.
12
1218
3892
0416
10
1C
, 1
1180
3923
0161
10
1H
and 16
1820
9283
1641
10
1O
The weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.
Exercise
1) Evaluate the determinant of the matrix.
a) i)
26
10 ii)
15
41 iii)
42
28 iv)
25
24 v)
31
08 vi)
12
39
vii)
27
117 viii)
43
04 ix)
93
56 x)
108
30 xi)
85
212
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b) i)
185
232-
1-412
ii)
110
124
495
iii)
145
41310
250
iv)
417
2420
2161
v)
250
980
104
vi)
4133
9312
928
Vii)
265
0910
1123
viii)
121511
18910
2023
ix)
1428
6010
10415
2) Find the area of the triangle with the given vertices
a) A(0, 1), B(2, 7), C(5, 5)
b) A(3, 6), B(3, 0), C(1, 3)
c) A(6, -1), B(2, 2), C(4, 8)
d) A(-4, 2), B(3, -1), C(-2, -2)
e) A(2, -6), B(-1, -4), C(0, 2)
f) A(1, 3), B(-2, 6), C(-1, 1)
3) Use Cramer’s rule to solve the linear system.
a) 454
486
yx
yx
b) 2383
372
yx
yx
c) 511114
2212
yx
yx
d) 465
32
yx
yx
e) 56103
1157
yx
yx
f) 4234
729
yx
yx
g) 1753
37
yx
yx
h) 511512
4412
yx
yx
i) 3478
1834
yx
yx
j) 2472
1354
yx
yx
k) 4075
3298
yx
yx
l) 641512
50103
yx
yx
4) Use Cramer’s rule to solve the linear system.
a)
4443
1
232
zyx
zyx
zyx
b)
4253
362
13
zyx
zyx
zyx
c)
23
86
10523
zy
zx
zyx
d)
125
3
92
zx
zyx
zyx
e)
9
17233
764
zyx
zyx
zyx
f)
2233
1522
74
zyx
zyx
zyx
g)
1656
924
52
yx
zyx
zyx
h)
14253
222
1372
zyx
zyx
zyx
i)
6458
829
1423
zyx
zyx
zyx
5) You are making a large pennant for your
school football team. A diagram of the
pennant is shown at the right. The
coordinates given are measured in
inches. How many square inches of
material will you need to make the
pennant?
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6) Black-necked stilts are birds that live
throughout Florida and surrounding
areas but breed mostly in the triangular
region shown on the map. Estimate the
area of this region. The coordinates
given are measured in miles.
7) On a Marconi-rigged sloop, there are two triangular sails, a mainsail and a jib. These sails are
shown in a coordinate plane at the right. The coordinates in the plane are measured in feet
a) Find the area of the mainsail shown.
b) Find the area of the jib shown.
c) Suppose you are making a scale model
of the sailboat with the sails shown using
a scale of l in. = 6 ft. What is the area of
the model’s mainsail?
8) You fill up your car with 10 gallons of premium gasoline and fill a small gas can with 2
gallons of regular gasoline for your lawn mower. You pay the cashier $13.56. The price of
premium gasoline is 12 cents more per gallon than the price of regular gasoline. Use a linear
system and Cramer’s rule to find the price per gallon for regular and premium gasoline
9) The Golden Triangle refers to a large
triangular region in India. The Taj Mahal
is one of the many wonders that lie
within the boundaries of this triangle.
The triangle is formed by imaginary
lines that connect the cities of New
Delhi, Jaipur, and Agra. Use the
coordinates on the map and a
determinant to estimate the area of the
Golden Triangle. The coordinates given
are measured in miles
10) The atomic weights of three compounds are shown
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Compound Formular Atomic weight
Tetrasulphur tetranitrate 44NS 184
Sulphur hexafluoride 6SF 146
Dinitrogen tetrafloride 42FN 104
Use a linear system and Cramer’s rule to find the atomic weights of sulphur (S), nitrogen
(N), and flourine (F).
11) Explain what happens to the determinant of a matrix when you switch two rows or two
columns.
Identity and Inverse Matrices The number 1 is the multiplicative identity for real numbers because aaa 11 . For
matrices, the nn identity matrix is the matrix that has 1’s on the main diagonal and 0’s
elsewhere.
Eg
10
01 and
100
010
001
are the 22 and 33 identity matrix respectively.
If A is any nn matrix and I is the nn identity matrix, then IA=AI=A.
Two nn matrices are inverses of each other if their product (in both orders) is the nn
identity matrix. For example, matrices A and B below are inverses of each other.
I10
01
33
12
23
13AB
and I
10
01
23
13
33
12BA
The symbol used for the inverse of A is -1A .
The Inverse of a 22 Matrix
The inverse of the matrix
dc
baA is
ac-
bd1
ac-
bd
A
1A 1-
bcadprovided
0bcad
Example 1 Find the inverse of
24
13A
Solution
5.12-
5.01
34-
12
46
1A 1- check the inverse by showing that AAIAA -1-1
10
01
24
13
5.12-
5.01AA and
10
01
5.12-
5.01
24
13AA 1-1-
Example 2 Given the matrices
13-
14A and
36-
58B Solve the matrix equation AX
= B for the 22 matrix X where
Solution
Begin by finding the inverse of A.
43
11
43
11
34
1A 1-
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To solve the equation for X, multiply both sides of the equation by -1A on the left.
30
22X
30
22X
10
011
36-
58
43
11X
13-
14
43
11BAAXA 1-1-
You can check the solution by multiplying A and X to see if you get B.
Some matrices do not have an inverse. You can tell whether a matrix has an inverse by
evaluating its determinant. If det A = 0, then A does not have an inverse. If det A ≠ 0, then A has
an inverse.
Exercise
1) Find the inverse of the matrix.
a)
23
34
b)
12
33
c)
46
01
d)
4
1
21
2
4
e)
45.2
35.0
f)
42.3
26.1
g)
43
54
h)
38
26
i)
171
181
j)
31
176
k)
13
27
l)
14_
27
m)
22
76
n)
44
45
o)
39
311
p)
12
2
1
2
3
q)
108
5.22.2
r)
2
5
4
3
5
4
1
2) Solve the matrix equation.
a)
04
13
50
135X
b)
2026
2017
28
15X
c)
513
604
10
42X
d)
1334
18512
14
35X
e)
92
37
151
58
41
73X
f)
66
91
34
43
54
97X
g)
11
23
03
12
64
21X
h)
28
64
75
11
26
34X
3) Tell whether the matrices are inverses of each other.
a)
103
31 and
13
310
b)
10121
5711
8102
and
437
325
120
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c)
354
120
243
and
618
314
8211
d)
221
052
543
and
2339
1014
25210
4) Use the matrices shown. The columns of matrix T give the coordinates of the vertices of a
triangle. Matrix A is a transformation matrix.
241
321T and
01
10A
a) Find AT and AAT. Then draw the original triangle and the two transformed triangles.
What transformation does A represent?
b) Suppose you start with the triangle determined by AAT and want to reverse the
transformation process to produce the triangle determined by AT and then the triangle
determined by T. Describe how you can do this.
Solving Systems Using Inverse Matrices
In session.3 you learned how to solve a system of linear equations using Cramer’s rule. Here you
will learn to solve a system using inverse matrices
To solve a system of linear equations using inverse we must first write the system as a matrix
equation AX = B, where the matrix A is the coefficient matrix of the system, X is the matrix of
variables, and B is the matrix of constants. For instance the system f
dycx
ebyax can be written
in matrix form as
f
e
y
x
dc
ba.
Once we have written a linear system as AX = B, the next step is to pre-multiply both sides of
the matrix equation by the inverse of the coefficient matrix and then simplify both sides. Ie
B A=X AXAB = AX 11 For instance
ce-af
f1
f
11
f
bde
bcady
xe
ac
bd
bcady
x
dc
ba
ac
bd
bcad
e
y
x
dc
ba
Finally obtain the solution by equating the corresponding entries of the resulting matrices in step
2 above. Ie bcad
ceay
bcad
bdex
-f and
f
Example 1 Use the inverse matrices to solve the linear system 102
543
yx
yx
Solution
Begin by writing the linear system in matrix form, ie
10
5
12
43
y
x
Then pre-multiply both sides by the inverse of the coefficient matrix to get
4
7
20
35
5
1
10
01
10
5
32
41
83
1
12
43
32
41
83
1
y
x
y
x
y
x
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 45
The solution of the system is (-7, -4). Check this solution in the original equations
Example 2 Show that matrices A and B are inverses of each other. Hence solve the system of
linear equations
242
133
132
zyx
zyx
zyx
where
142
133
132
A and
326
101
011
B
Solution
Start by showing that AB = BA = I
142
133
132
326
101
011
100
010
001
326
101
011
142
133
132
write the linear system in matrix form then pre-multiply both sides by the inverse to get
2
1
2
2
1
1
326
101
011
142
133
132
326
101
011
2
1
1
142
133
132
z
y
x
z
y
x
z
y
x
Thus the solution of the system is (2, -1, -2).
Exercise
1) Write the linear system as a matrix equation.
a) 62
8
yx
yx
b) 724
93
yx
yx
c) 843
5
yx
yx
d) 54
62
yx
yx
e) 1024
935
yx
yx
f) 1573
1152
yx
yx
g) 1154
48
yx
yx
h) 13
452
yx
yx
i)
843
15
510
zyx
yx
zyx
j)
38254
2372
454
zyx
zyx
zyx
k)
313
1042
1643
zyx
zyx
zyx
l)
8.43.48.43.0
2.27.05.22.1
9.52.01.35.0
zyx
zyx
zyx
m)
472
62
9
zyx
zyx
zx
n)
059
14126
23108
zx
zy
zy
o)
2
12
0
zy
zx
zyx
2) Use an inverse matrix to solve the linear system.
a) 2187
2
yx
yx
b) 182
3
yx
yx
c) 1026
634
yx
yx
d) 1125
823
yx
yx
e) 81211
1
yx
yx
f) 223
5372
yx
yx
g) 434
857
yx
yx
h) 3042
5475
yx
yx
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 46
i) 332
975
yx
yx
j) 1432
92
yx
yx
k) 3152
2642
yx
yx
l) 2222
4359
yx
yx
3) Use the given inverse of the coefficient matrix to solve the linear system.
a)
5437
4325
22
zyx
zyx
zy
10141
571
8111
A 1-
b)
43
3025
93
yx
zyx
zyx
741
1693
531
A 1-
4) You are planning a birthday party for your younger brother at a skating rink. The cost of
admission is $3.50 per adult and $2.25 per child, and there is a limit of 20 people. You have
$50 to spend. Use an inverse matrix to determine how many adults and how many children
you can invite.
5) The price of flatware varies depending on the number of place settings you buy as well as
other items included in the set. Suppose a set with 4 place settings costs $142 and a set with 8
place settings and a serving set costs $351. Find the cost of a place setting and a serving set.
Assume that the cost of each item is the same for each flatware set.
6) Solve the linear system using the given inverse of the coefficient matrix.
6326
3335
5272
2336
zyx
zyxw
zyxw
zyxw
520224
833339
0101
933340
A 1-
7) Solve the system of linear equations using any algebraic method.
a) 823
64
yx
yx
b) 453
92
yx
yx
c) 223
159
yx
yx
d) 1
822
yx
yx
e) 726
143
yx
yx
f) 4083
1025
yx
yx
g) 3252
1137
yx
yx
h) 1092
145
yx
yx
i) 2412
497
yx
yx
932
1323
1323
zyx
zyx
yx
j)
7522
14823
43
zyx
zyx
zyx
k)
16652
1584
21553
zyx
zyx
zyx
l)
022
55
22
zyx
zyx
zx
m)
21353
96
1434
zyx
yx
zyx
n)
1127
122
173
zyx
zx
zyx
8) Dentists use various amalgams for silver fillings. The matrix shows the percents (expressed
as decimals) of powdered alloys used in preparing three different amalgams. Suppose a
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 47
dentist has 5483 grams of silver, 2009 grams of tin, and 129 grams of copper. How much of
each amalgam can be made?
PERCENT ALLOY BY WEIGHT
Amalgam
copper
Tin
Silver
00.003.004.0
27.025.026.0
73.072.070.0
C BA
9) You are making mosaic tiles from three types of stained glass. You need 6 square feet of
glass for the project and you want there to be as much iridescent glass as red and blue glass
combined. The cost of a sheet of glass having an area of 0.75 square foot is $6.50 for
iridescent, $4.50 for red, and $5.50 for blue. How many sheets of each type should you
purchase if you plan to spend $45 on the project?
10) You are an accountant for a construction business and are planning next year’s budget. You
have $200,000 to spend on salaries, equipment maintenance, and other general expenses.
Based on previous financial records of the business, you expect to spend five times as much
on salaries as on equipment maintenance, and you expect general expenses to be 10% of the
amount spent on the other two categories combined. Write and solve a system of equations to
find the amount you should budget for each category.
11) A company sells different sizes of gift baskets with a varying assortment of meat and cheese.
A basic basket with 2 cheeses and 3 meats costs $15, a big basket with 3 cheeses and 5 meats
costs $24, and a super basket with 7 cheeses and 10 meats costs $50.
a) Write and solve a system of equations using the information about the basic and big
baskets.
b) Write and solve a system of equations using the information about the big and super
baskets.
c) Compare the results from parts (a) and (b) and make a conjecture about why there is a
discrepancy.
12) A walkway lighting package includes a transformer, a certain length of wire, and a certain
number of lights on the wire. The price of each lighting package depends on the length of
wire and the number of lights on the wire.
• A package that contains a transformer, 25 feet of wire, and 5 lights costs $20.
• A package that contains a transformer, 50 feet of wire, and 15 lights costs $35.
• A package that contains a transformer, 100 feet of wire, and 20 lights costs $50.
Write and solve a system of equations to find the cost of a transformer, the cost per foot of
wire, and the cost of a light. Assume the cost of each item is the same in each lighting
package.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 48
Permutations, Combinations and Binomial Expansion
The Fundamental Counting Principle and Permutations
In many real-life problems you want to count the number of possibilities. For instance, suppose
you own a small deli. You offer 4 types of meat (ham, turkey, roast beef, and pastrami) and 3
types of bread (white, wheat, and rye). How many choices do your customers have for a meat
sandwich?
To count the number of possible sandwiches we can use the fundamental counting principle.
Because you have 4 choices for meat and 3 choices for bread, the total number of choices is
1234 .
The Fundamental Counting Principle
Two Events If one event can occur in m ways and another event can occur in n ways, then the
number of ways that both events can occur is nm . For instance, if one event can occur in 12
ways and another event can occur in 5 ways, then both events can occur in 60512 ways.
Three or More Events. The fundamental counting principle can be extended to three or more
events. For example, if three events can occur in m, n, and p ways, then the number of ways that
all three events can occur is pnm . For instance, if three events can occur in 3, 5, and 4 ways,
then all three events can occur in 60453 ways.
Example 1 Police use photographs of various facial features to help witnesses identify suspects.
One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths,
and 74 chins and cheeks.
a) The developer of the identification kit claims that it can produce billions of different faces. Is
this claim correct?
b) A witness can clearly remember the hairline and the eyes and eyebrows of a suspect. How
many different faces can be produced with this information?
Solution
a) You can use the fundamental counting principle to find the total number of different faces.
Number of faces = 195 × 99 × 89 × 105 × 74 = 13,349,986,650 The developer’s claim
is correct since the kit can produce over 13 billion faces
b) Because the witness clearly remembers the hairline and the eyes and eyebrows, there is only
1 choice for each of these features. You can use the fundamental counting principle to find
the number of different faces. Number of faces = 1 × 1 × 89 × 105 × 74 = 691,530 The
number of faces that can be produced has been reduced to 691,530.
Example 2 The standard configuration for a New York license plate is 3 digits followed by 3
letters.
a) How many different license plates are possible if digits and letters can be repeated?
b) How many different license plates are possible if digits and letters cannot be repeated?
Solution
a) There are 10 choices for each digit and 26 choices for each letter. You can use the
fundamental counting principle to find the number of different plates.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 49
Number of plates = 10 × 10 × 10 × 26 × 26 × 26 = 17,576,000
The number of different license plates is 17,576,000.
b) If you cannot repeat digits there are still 10 choices for the first digit, but then only 9
remaining choices for the second digit and only 8 remaining choices for the third digit.
Similarly, there are 26 choices for the first letter, 25 choices for the second letter, and 24
choices for the third letter. You can use the fundamental counting principle to find the
number of different plates. Number of plates = 10 • 9 • 8 • 26 • 25 • 24 = 11,232,000 The
number of different license plates is 11,232,000.
Permutations
Definition (n factorial) We define n!, pronounced n factorial as 123...)2)(1(! nnnn
Eg 6123!3 241234!4 12012345!5 720123456!6
Permutation; number of different arrangements of a group of items where order matters
When order matters BAAB .
An ordering of n objects is a Permutations of the objects. For instance, there are six
permutations of the letters A, B, and C that is ABC, ACB, BAC, BCA, CAB, CBA. The
fundamental counting principle can be used to determine the number of permutations of n
objects. For instance, you can find the number of ways you can arrange the letters A, B, and C by
multiplying. There are 3 choices for the first letter, 2 choices for the second letter, and 1 choice
for the third letter, so there are 6123 ways to arrange the letters.
In general, the number of permutations of n distinct objects is: 123...)2)(1(! nnnn
and the number of permutations of n objects taken r at a time is: )!(
!
rn
nrn p
Example 3 Twelve skiers are competing in the final round of the Olympic freestyle skiing
aerial competition.
a) In how many different ways can the skiers finish the competition? (Assume there are no ties.)
b) In how many different ways can 3 of the skiers finish first, second, and third to win the gold,
silver, and bronze medals?
Solution
a) There are 12! different ways that the skiers can finish the competition. 12! = 479,001,600
b) Any of the 12 skiers can finish first, then any of the remaining 11 skiers can finish second,
and finally any of the remaining 10 skiers can finish third. So, the number of ways that the
skiers can win the medals is: 12 × 11 × 10 = 1320 This is the number of permutations of 12
objects taken 3 at a time, is denoted by 312 p , and is given by )!512(
!12
Example 4 You are considering 10 different colleges. Before you decide to apply to the
colleges, you want to visit some or all of them. In how many orders can you visit (a) 6 of the
colleges and (b) all 10 colleges?
Solution
a) The number of permutations of 10 objects taken 6 at a time is 200,151610 p
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Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 50
b) The number of permutations of 10 objects taken 10 at a time is: 800,628,31010 p
So far you have been finding permutations of distinct objects (ie we dealt with selections without
repetitions). If some of the objects are repeated, then some of the permutations are not
distinguishable. For instance, of the six ways to order the letters M, O, and M
M O M M O M O M M O M M M M O M M O
Only three are distinguishable without color: MOM, OMM, and MMO. In this case, the number
of permutations is 3!2
!3 , not 3!=6
The number of distinguishable permutations of n objects where 1r are identical objects of type
1, 2r are identical objects of type 2,. . . . kr are identical objects of type k. is given by
!.....!!
!
21 krrr
n
Example 4 Find the number of distinguishable permutations of the letters in (a) OHIO and (b)
MISSISSIPPI.
Solution
a) OHIO has 4 letters of which O is repeated 2 times. So, the number of distinguishable
permutations is 12!2
!4 .
b) MISSISSIPPI has 11 letters of which I is repeated 4 times, S is repeated 4 times, and P is
repeated 2 times. So, the number of distinguishable permutations is 650,34!2!4!4
!11
Exercise
1. Find the number of possible outcomes if you toss a coin and roll a dice.
2. Find the number of distinguishable permutations of the letters in the word a) FORMAT b)
WYOMING c) THURSDAY d) SEPTEMBER e) MATHEMATICS
3. How many 3-digit numbers are possible using the digits 0, 1, 4, 5, 7 and 9?
4. How many arrangements of three types of flowers are there if there are 6 types to choose
from?
5. How many different ways can you have 1st 2nd 3rd and 4th in a race with 10 runners?
6. How many ways can 7 people be arranged around a roundtable?
7. A license plate has 3 letters and 3 digits in that order. A witness to a hit and run accident saw
the first 2 letters and the last digit. If the letters and digits can be repeated, how many license
plates must be checked by the police to find the culprit?
8. You are going to set up a stereo system by purchasing separate components. In your price
range you find 5 different receivers, 8 different compact disc players, and 12 different
speaker systems. If you want one of each of these components, how many different stereo
systems are possible?
9. A pizza shop runs a special where you can buy a large pizza with one cheese, one vegetable,
and one meat for $9.00. You have a choice of 7 cheeses, 11 vegetables, and 6 meats.
Additionally, you have a choice of 3 crusts and 2 sauces. How many different variations of
the pizza special are possible?
10. To keep computer files secure, many programs require the user to enter a password. The
shortest allowable passwords are typically six characters long and can contain both numbers
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Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 51
and letters. How many six-character passwords are possible if (a) characters can be repeated
and (b) characters cannot be repeated?
11. Simplify the formula for nPr when r = 0. Explain why this result makes sense.
12. A particular classroom has 24 seats and 24 students. Assuming the seats are not moved, how many
different seating arrangements are possible? Write your answer in scientific notation.
13. “Ringing the changes” is a process where the bells in a tower are rung in all possible permutations.
Westminster Abbey has 10 bells in its tower. In how many ways can its bells be rung?
14. A music store wants to display 3 identical keyboards, 2 identical trumpets, and 2 identical guitars in
its store window. How many distinguishable displays are possible?
15. In a dog show how many ways can 3 Chihuahuas, 5 Labradors, 4 poodles, and 3 beagles line up in
front of the judges if the dogs of the same breed are considered identical?
16. Find the number of permutations of n objects taken n º 1 at a time for any positive integer n. Compare
this answer with the number of permutations of all n objects. Does this make sense? Explain
17. You have learned that n! represents the number of ways that n objects can be placed in a linear order,
where it matters which object is placed first. Now consider circular permutations, where
objects are placed in a circle so it does not matter which object is placed first. Find a formula
for the number of permutations of n objects placed in clockwise order around a circle when
only the relative order of the objects matters. Explain how you derived your formula.
Combinations and the Binomial Theorem
Combination; different arrangements of a group of items where order does not matter
When order is not important BAAB .
The number of combinations of a group of n distinct objects taken r at a time is: )!(!
!
rnr
nrnC
for
instance, How many ways are there to select a subcommittee of 7 members from among a
committee of 17? Ans 448,19717 C
Example 1 A standard deck of 52 playing
cards has 4 suits with 13 different cards in
each suit as shown. a. If the order in which
the cards are dealt is not important, how
many different 5-card hands are possible? b.
In how many of these hands are all five
cards of the same suit?
Solution
a) The number of ways to choose 5 cards
from a deck of 52 cards is:
960,598,2552 C
b) For all five cards to be the same suit, you
need to choose 1 of the 4 suits and then 5
of the 13 cards in the suit. So, the
number of possible hands is:
148,551314 CC
When finding the number of ways both an event A and an event B can occur, you need to
multiply (as you did in part (b) of Example 1). When finding the number of ways that an event A
or an event B can occur, you add instead.
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Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 52
Example 2 A restaurant serves omelets that can be ordered with any of the ingredients shown.
a) Suppose you want exactly 2 vegetarian
ingredients and 1 meat ingredient in your
omelet. How many different types of
omelets can you order?
b) Suppose you can afford at most 3
ingredients in your omelet. How many
different types of omelets can you order?
Solution
a) You can choose 2 of 6 vegetarian ingredients and 1 of 4 meat ingredients. So, the number of
possible omelets is: 601426 CC
b) You can order an omelet with 0, 1, 2, or 3 ingredients. Because there are 10 items to choose
from, the number of possible omelets is: 17612045101310210110010 CCCC
Remark Counting problems that involve phrases like “at least” or “at most” are sometimes
easier to solve by subtracting possibilities you do not want from the total number of possibilities.
Example 3 A theater is staging a series of 12 different plays. You want to attend at least 3 of
the plays. How many different combinations of plays can you attend?
Solution
You want to attend 3 plays, or 4 plays, or 5 plays, and so on. So, the number of combinations of
plays you can attend is 1212512412312 ... CCCC . Instead of adding these combinations, it is
easier to use the following reasoning. For each of the 12 plays, you can choose to attend or not
attend the play, so there are 122 total combinations. If you attend at least 3 plays you do not
attend only 0, 1, or 2 plays. So, the number of ways you can attend at least 3 plays is:
017,4)66121(086,4)(2 212112012
12 CCC
Binomial Theorem
If you arrange the values of nCr in a triangular pattern in which each row corresponds to a value
of n, you get what is called Pascal’s triangle. It is named after the famous French mathematician
Blaise Pascal (1623–1662)
Pascal’s triangle has many interesting patterns and properties. For instance, each number other
than 1 is the sum of the two numbers directly above it.
The binomial expansion of nqp )( for any positive integer n is:
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n
r
rrn
rn
n
nn
n
n
n
n
n
n
n qpCqpCqpCqpCqpCqp0
022
2
11
1
0
0 .....)(
To expand a power of a binomial difference, you can rewrite the binomial as a sum. The
resulting expansion will have terms whose signs alternate between + and º.
Example 1 Expand a) 4)2( x b)
32 )( yx c) 5)( yx d)
4)25( x
Solution
1632248
22222)2(
234
40
44
31
34
22
24
13
14
04
04
4
xxxx
xCxCxCxCxCx
64223
320
33
221
23
122
13
023
03
32
33
)()()()()(
yxyyxx
yxCyxCyxCyxCyx
54322345
50
55
41
45
32
35
23
25
14
15
05
05
5
510105
)()()()()()()(
yxyyxyxyxx
yxCyxCyxCyxCyxCyxCyx
432
40
44
31
34
22
24
13
14
04
04
4
161606001000625
)2(5)2(5)2(5)2(5)2(5)25(
xxxx
xCxCxCxCxCx
Example 2 Find the coefficient of 4x in the expansion of 12)32( x
Solution
From the binomial theorem you know the following:
12
0
12
12
12 )3()2()32(r
rr
r xCx
The term that has 4x occurs when r=8 and is is 4484
812 120,963,51656116495)3()2( xxxC
The coefficient is 51,963,120.
Exercise
1) Your English teacher has asked you to select 3 novels from a list of 10 to read as an
independent project. In how many ways can you choose which books to read?
2) Your friend is having a party and has 15 games to choose from. There is enough time to play
4 games. In how many ways can you choose which games to play?
3) You are buying a new car. There are 7 different colors to choose from and 10 different types
of optional equipment you can buy. You can choose only 1 color for your car and can afford
only 2 of the options. How many combinations are there for your car?
4) There are 6 artists each presenting 5 works of art in an art contest. The 4 works judged best
will be displayed in a local gallery. In how many ways can these 4 works all be chosen from
the same artist’s collection?
5) An amusement park has 20 different rides. You want to ride at least 15 of them. How many
different combinations of rides can you go on?
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 54
6) A summer concert series has 12 different performing artists. You decide to attend at least 4
of the concerts. How many different combinations of concerts can you attend? Decide
whether the problem requires combinations or permutations to find the answer. Then solve
the problem.
7) You are buying a flower arrangement. The florist has 12 types of flowers and 6 types of
vases. If you can afford exactly 3 types of flowers and need only 1 vase, how many different
arrangements can you buy?
8) Eight members of a school marching band are auditioning for 3 drum major positions. In how
many ways can students be chosen to be drum majors?
9) Your school yearbook has an editor-in-chief and an assistant editor-in-chief. The staff of the
yearbook has 15 students. In how many ways can students be chosen for these 2 positions?
10) A relay race has 4 runners who run different parts of the race. There are 16 students on your
track team. In how many ways can your coach select students to compete in the race? 58.
11) You must take 6 elective classes to meet your graduation requirements for college. There are
12 classes that you are interested in. In how many ways can you select your elective classes?
12) A group of 20 high school students is volunteering to help elderly members of their
community. Each student will be assigned a job based on requests received for help. There
are 8 requests for raking leaves, 7 requests for running errands, and 5 requests for washing
windows. a. One way to count the number of possible job assignments is to find the number
of permutations of 8 L’s (for “leaves”), 7 E’s (for “errands”), and 5 W’s (for “windows”).
a) Use this method to write the number of possible job assignments first as an
expression involving factorials and then as a simple number.
b) Another way to count the number of possible job assignments is to first choose the 8
students who will rake leaves, then choose the 7 students who will run errands from
the students who remain, and then choose the 5 students who will wash windows
from the students who still remain. Use this method to write the number of possible
job assignments first as an expression involving factorials and then as a simple
number.
c) Writing How do the answers to parts (a) and (b) compare to each other? Explain why
this makes sense.
13) Expand a) 6)4( x b)
6)3( yx c) 72 )( yx d)
73)2( yx
14) Use the binomial theorem to write the binomial expansion. a) 5)3( yx b)
6)2( yx c)
53 )3( x d) 72 )2( yx d)
4)33( x e)72 )2( yx f)
423 )( yx
15) Find the coefficient of 6x in the expansion of; a) 8)2( x b)
12
2
1
xx c)
10
2
3 3
2
x
x. Also
find the constant term for part c and d.
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Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 55
Functions and Their Graphs
Relations and Functions In natural language relations are a kind of links existing between objects. Examples: ‘mother of’,
‘neighbor of’, “part of”, ‘is older than’, ‘is an ancestor of’, ‘is a subset of’, etc. These are binary
relations. Mathematically A relation is a mapping, or pairing, of input values with output
values. The set of input values is called the domain and the set of output values is called the
range.
Definition A function f from a set X to a set Y is a rule that assigns to each element of X a
unique element of Y . We could write f ∶ 𝑎 → 𝑏 to indicate that f sends a to b. But the most
common notation (called the function notation) is f(𝑎) = 𝑏: read as “the value of ƒ at x,” or
simply as “ƒ of x.” This notation means that when the input a is inserted into the function f, the
output is b.
The set X of inputs is called the domain of the function f. The set Y of all conceivable outputs is
the codomain of the function f. The set of all outputs is the range of f. The range is a subset of Y.
A relation is a function provided there is exactly one output for each input. It is not a function if
at least one input has more than one output.
Relations (and functions) between two quantities can be represented in many ways, including
mapping diagrams, tables, graphs, equations, and verbal descriptions
Example 1 Identify the domain and range. Then tell whether the relation is a function.
Solution
a. The domain consists of -3, 1, and 4, and
the range consists of -2, 1, 3, and 4.
The relation is not a function because the
input 1 is mapped onto both -2 and 1.
b. The domain consists of -3, 1, 3, and 4,
and the range consists of -2, 1, and 3. The relation is a function; each input in the domain is mapped onto exactly one output in the range.
A relation can be represented by a set of ordered pairs of the form (x, y). In an ordered pair the
first number is the x-coordinate and the second number is the y-coordinate. To graph a relation,
plot each of its ordered pairs in a coordinate plane such as the one shown. A coordinate plane is
divided into four quadrants by the x-axis and the y-axis. The axes intersect at a point called the
origin.
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Example 2 Graph the relations given in Example 1.
Solution
a) Write the relation as a set o ordered
pairs: (-3, 3), (1, -2), (1, 1), (4, 4). Then
plot the points in a coordinate plane.
b) Write the relation as a set of ordered
pairs: (-3, 3), (1, 1), (3, 1), (4, -2). Then
plot the points in a coordinate plane.
In Example 2 notice that the graph of the relation that is not a function (the graph on the left) has
two points that lie on the same vertical line. You can use this property as a graphical test for
functions.
Vertical Line Test for Functions: A relation is a function if and only if no vertical line
intersects the graph of the relation at more than one point.
Variables other than x and y are often used when working with relations in real-life situations, as
shown in the next example.
Example 3 The graph shows the ages a
and diameters d of several pine trees at Lund
breck Falls in Canada. Are the diameters of
the trees a function of their ages? Explain.
Solution
The diameters of the trees are not a function
of their ages because there is at least one
vertical line that intersects the graph at more
than one point. For example, a vertical line
intersects the graph at the points (75, 1.22)
and (75, 1.58). So, at least two trees have the
same age but different diameters.
Graphing and Evaluating Functions Many functions can be represented by an equation in two variables, such as 72 xy . An
ordered pair (x, y) is a solution of such an equation if the equation is true when the values of x
and y are substituted into the equation. For instance, (2, 3) is a solution of 72 xy because
72 xy is a true statement.
In an equation, the input variable is called the independent variable. The output variable is
called the dependent variable and depends on the value of the input variable. For the equation y
= 2x-7, the independent variable is x and the dependent variable is y.
The graph of an equation in two variables is the collection of all points (x, y) whose coordinates
are solutions of the equation.
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Graphing Equations in Two Variables
To graph an equation in two variables, follow these steps:
i) Construct a table of values. (ii) Graph enough solutions to recognize a pattern.
(iii) Connect the points with a line or a curve.
Example 3 Graph the function y = x + 1.
Solution
Begin by constructing a table of values
Choose x. -2 -1 0 1 2
Evaluate y -1 0 1 2 3
Plot the points. Notice the five points lie on
a line.
Draw a line through the points
The function in Example 4 is linear function a linear function because it is of the form
cmxy where m and c are constants. The graph of a linear function is a line. By naming a
function “ƒ” you can write the function using function notation f(𝑥) = 𝑚𝑥 + 𝑐
The symbol ƒ(x) is read as “the value of ƒ at x,” or simply as “ƒ of x.” Note that ƒ(x) is another
name for y. The domain of a function consists of the values of x for which the function is
defined. The range consists of the values of ƒ(x) where x is in the domain of ƒ. Functions do not
have to be represented by the letter ƒ. Other letters such as g or h can also be used.
Example 5 Decide whether the function is linear. Then evaluate the function when x =-2.
a) 53)f( 2 xxx b) 62)( xxg
Solution
a) f(x) is not a linear function because it has an 2x -term. Now 55)2(3)2()2f( 2
b) g(x) is a linear function because it has the form g(x) = mx + b. Now 26)2(2)2( g
Example 6 In March of 1999, Bertrand Piccard and Brian Jones attempted to become the first
people to fly around the world in a balloon. Based on an average speed of 97.8 kilometers per
hour, the distance d (in kilometers) that they traveled can be modeled by d = 97.8t where t is the
time (in hours). They traveled a total of about 478 hours. The rules governing the record state
that the minimum distance covered must be at least 26,700 kilometers
a) Identify the domain and range and
determine whether Piccard and Jones set
the record.
b) Graph the function. Then use the graph
to approximate how long it took them to
travel 20,000 kilometers.
Solution
a) Because their trip lasted 478 hours, the
domain is 0 ≤ t ≤ 478. The distance they
traveled was 700,46)478(8.97 d km,
so the range is 0 ≤ d ≤ 46,700. Since
46,700 > 26,700, they did set the record
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b) The graph of the function is shown. Note that the graph ends at (478, 46,700). To find how
long it took them to travel 20,000 kilometers, start at 20,000 on the d-axis and move right
until you reach the graph. Then move down to the t-axis. It took them about 200 hours to
travel 20,000 kilometers
Functions defined by equations
Most of the functions that we will explore in mathematics and science are defined by some
equation. For example, consider the function f ℝ → ℝ defined by mapping a real number to the
number that is one more than its square, that is f(𝑥) = 𝑥2 + 1
This function maps real numbers (the set ℝ) to real numbers and so the domain of f is ℝ and the
codomain of f is ℝ. However, the range of f is the set of all real numbers greater than or equal to
1. In interval notation, the domain of f is ),( and the range is ),1[ .
We can often define a function implicitly in an equation involving two variables. Traditionally
we use the letter x for the input variable and the letter y for the output variable. The equation
𝑥2 − 𝑦 = −1 defines the function f(𝑥) = 𝑥2 + 1 discussed above.
As another example, consider the linear equation 0432 yx
This equation can be viewed as defining a function with inputs x and outputs y. From this
viewpoint, we can solve for y and get )24(3
1 xy
We might then explicitly define the function )24()(f3
1 xx
Note: Our choice of x as input and y as output was somewhat arbitrary. We could have decided
(Contrary to custom) that y is the input and x is the output. Then, solving for x, we have
)34(2
1 yx and so we can create the function x = g(y) (with input y and output x)
)34()(g2
1 yy
Example 2 Does the equation 𝑥2𝑦 = 4 define y as a function of x? (If it does, give the domain
of the implied function.)
Solution. We attempt to solve for y. We may divide both sides of the equation by 𝑥2 as long as x
is not zero. This gives us 2
4
xy :
Is there a problem with x = 0? No, x = 0 does not allow 𝑥2𝑦 = 4, so x will never be zero in this
equation. YES; 2
4
xy :
The domain of this function is all real numbers except zero. In interval notation this is
),0()0,(
Example 3 Does the equation 42 xy define y as a function of x?
Solution.
If we attempt to solve for y, we divide both sides of the equation by x (as long as 0x ) and so
we have xx
yy 442 ie y could be positive or negative { there will generally be two
choices here.
NO; for example, if x = 1 then we don't know if y = 2 or y = -2:
Example 4 Does the equation 02 yx define y as a function of x? (Why/why not?)
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Solution.
Although it might be tempting to solve for y, first notice that if x is zero then y could be 0 or 1 or
2.71828 or anything! So the input x = 0 does not give a unique output. This is not a function.
NO; if x = 0 then y could be anything.
(Note how this is different than example 2. In example 2, x = 0 is not a possible input in the
equation. But here x = 0 is a possibility for a solution to the equation!)
Let us practice the function notation, f(x).A formula for f(x) tells us how the input x leads to the
output f(x):
Example 5 Suppose 9)(f 2 xx . Compute: f(0), f(1), f(-1), f(-5), f(-x) , f(x + h), f( x ),
f(2a + 1) and -f(x) + 2
Solution
If 9)(f 2 xx then 990)0(f 2 891)1(f 2 , 89)1()1(f 2
169)5()5(f 2 99)()(f 22 xxx 929)()(f 222 hxhxhxhx
99)()(f 2 xxx 8449)12()12(f 22 aaaa 22 112)9(2)(f- xxx
Question: Apply the mapping g: 𝑥 → 3𝑥 to the domain {0, 2, 4, 6, 8}. List the image set
Given that f: 𝑥 →1
𝑥 and g: 𝑥 → 1 − 𝑥 write down the values of f(1) f(0) and f(−𝑥)
The domain of a function
The domain of a function is generally viewed as the largest possible set of inputs into the
function. For example, the domain of the function xx )(f is all real numbers greater than or
equal to zero. In interval notation we write ),0[ . Note that we cannot evaluate f(x) at negative
numbers (if we assume we are always working with real numbers.)
We often need to find the domain of a function and if the inputs are real numbers, express the
domain in interval notation. When we do this, it is often easier to ask the question, \What is not
in the domain?"
For example, in the square root function, xx )(f we might ask the question, \Which numbers
do not have a square root?" Since the square of a real number cannot be negative, then our
answer is \We cannot take the square root of negative numbers." So the domain must be numbers
which are not negative, that is, zero and positive real numbers. So the domain of xx )(f is
),0[ : (We can indeed take the square root of 0 so we want to include 0 in the domain.)
Example 6 Find the domain of the function 512
32
2
1)(g
x
x
x
xx
Solution
What numbers cannot serve as input to g(x)? Since we cannot have denominators equal to zero
then 2x cannot be an input; neither can 2
1x : So the domain of this function g is all real
numbers except 2x and 2
1x .
There are several ways to write the domain of g. Using set notation, we could write the domain
as {𝑥𝜖ℝ: 𝑥 ≠ −2 , −1
2} This is a precise symbolic way to say, “All real numbers except -2 &
21 "
We could also write the domain in interval notation: ),(),2()2,(2
1
2
1
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This notation says that the domain includes all the real numbers smaller than -2, along with all
the real numbers between -2 and 21 , along with (in addition) the real numbers larger than
21 :
Example 7 Find the domain of the function;
a) xx 2)(f b) 1)(f xx c) 3
1)(f
x
xx d)
86
1)(f
2
xx
xx
Solution
a) Since the square root function requires nonnegative inputs, we must have 02 x :
Add x to both sides of the inequality to get x2 : In interval notation this is ]2,(
Answer: The domain is ]2,(
b) Since the square root function requires nonnegative inputs, we must have 01x : If we add
1 to both sides of the inequality we have 1x :
Answer: The domain is ),1[
c) Again, as in the part b, we must have 1x but we must also prevent the denominator from
being zero, so x cannot be 3, either.
Answer: The domain is ),3()3,1[
d) We must have 1x and we must prevent the denominator from being zero. The denominator
factors as )4)(2(862 xxxx , so x cannot be 2 or 4. So our answer is all real numbers
at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is:
),4()4,2()2,1[ .
The domain is ),4()4,2()2,1[
Exercise
1) Is a function always a relation? Is a relation always a function? Explain your reasoning.
2) Explain why a vertical line, rather than a horizontal line, is used to determine if a graph
represents a function.
3) Decide whether the function is linear. Then evaluate the function for the given value of x.
a) )4(f;11)(f xx b) )4(f;21)(f x c) )6(f;5)(f xx
d) )(f;43)(f2
1 xx e) )2(f;29)(f 23 xxx f) )6(f;5)(f 2
3
2 xxx
4) The volume of a cube with side length s is given by the function 3)(v ss . Find )(v2
3 ).
Explain what )(v2
3 represents.
5) The volume of a sphere with radius r is given by the function 3
3
4)(v rr . Find V )2(v ).
Explain what V(2) represents.
6) Identify the domain and range of the
relation shown. Then tell whether the
relation is a function.
7) Graph the function a xy 4 b) 1 xy
b) 2
1
2
3 xy c) xy 9 d) xy3
24
8) Evaluate the function when x = 3
a) xx 6)(f b) 72)(f xx c) 2)(f xx d) 210)(f xx e) xxx 7)(f 3
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9) A car has a 16 gallon gas tank. On a long highway trip, gas is used at a rate of about 2 gallons
per hour. The gallons of gas g in the car’s tank can be modeled by the equation 𝑔 = 16 − 2𝑡
where t is the time (in hours).
a) Identify the domain and range of the function. Then graph the function.
b) At the end of the trip there are 2 gallons of gas left. How long was the trip?
10) Identify the domain and range.
11) Graph the relation. Then tell whether the
relation is a function
x 0 0 2 2 4 4
y 4 -4 -3 3 -1 1
x -5 -4 -3 0 3 4 5
y -6 -4 -2 -1 -2 -4 -6
x -2 -2 0 2 2
y 1.5 -3.5 0 1.5 -3.5
12) Use a mapping diagram to represent the relation. Then tell whether the relation is a function.
13) Use the vertical line test to determine whether the relation is a function.
14) Why does y = 3 represent a function, but
x = 3 does not?
15) The graph shows the ages and finishing
places of the top three competitors in
each of the four categories of the 100th
Boston Marathon. Is the finishing place
of a competitor a function of his or her
age? Explain your reasoning.
16) Evaluate the expression for the given values of x and y.
a) 9
6
x
y when x = -3 and y = -2
b) 9
11
x
y when x = -4 and y = 5
c) 3
)5(
x
y when x = 2 and y = 5
d) )4(
)1(
x
y when x = 6 and y = 4
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e) x
y
1
4 when x = 2 and y = 3 f)
x
y
14
10 when x = 6 and y = 8
17) The table below shows the number of shots attempted and the number of shots made by 9
members of the Utah Jazz basketball team in Game 1 of the 1998 NBA Finals.
Player Shots
attempted, x
Shots
made, y
Bryon Russell 12 6
Karl Malone 25 9
Greg Foster 5 1
Jeff Hornacek 10 2
John Stockton 12 9
Howard Eisley 6 4
Chris Morris 6 3
Greg Ostertag 1 1
Shandon Anderson 5 3
a) Identify the domain and range of the
relation. Then graph the relation.
b) Is the relation a function? Explain.
18) Water pressure can be measured in
atmospheres, where 1 atmosphere equals
14.7 pounds per square inch. At sea level
the water pressure is 1 atmosphere, and
it increases by 1 atmosphere for every 33
feet in depth. Therefore, the water
pressure p can be modeled as a function
of the depth d by this equation:
1300,133
1 ddp
a) Identify the domain and range of the
function. Then graph the function.
b) What is the water pressure at a depth
of 100 feet?
19) The graph shows the number of Independent representatives for the 100th–105th Congresses.
Is the number of Independent representatives a function of the Congress number? Explain
your reasoning.
20) Your cap size is based on your head
circumference (in inches). For head
circumferences from 8
720 inches to 25
inches, cap size s can be modeled as a
function of head circumference c by this
equation: )1(33
1 cs
a) Identify the domain and range of the
function. Then graph the function.
b) If you wear a size 7 cap, what is your head circumference?
21) For the numbers 2 through 9 on a
telephone keypad, draw two mapping
diagrams: one mapping numbers onto
letters, and the other mapping letters
onto numbers. Are both relations
functions? Explain.
Composite Functions
Composition of functions is when one function is inside another function. For example, if we
look at the function 2)12()(h xx . We can say that this function, h(x), was formed by the
composition of two other functions, the inside function 2𝑥 − 1 and the outside function z2. The
letter z was used just to represent a different variable, we could have used any letter that we
wanted. Notice that if we put the inside function, 2𝑥 − 1, into the outside function, z2, we would
get 𝑧2 = (2𝑥 − 1)2, which is our original function h(x).
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The notation used for the composition of functions looks like this, )(g)(f x . So what does this
mean? )(g)(f x the composition of the function f with g is defined as )(gf)(g)(f xx ,
notice that in the case the function g is inside the function f.
In composite functions it is very important that we pay close attention to the order in which the
composition of the functions is written. In many cases )(g)(f x is not the same as )(fg x
)(gf)(g)(f xx , the g function is inside the f function
)(fg)(f)(g xx the f function is inside of the g function
Therefore )(g)(f x and )(f)(g x are often different because in the composite )(g)(f x , f(x) is
the outside function and g(x) is the inside function. Whereas in the composite )(f)(g x , g(x) is
the outside function and f(x) is the inside function.
Example 1 If 4x-9)(f x and 7-2x)(g x , find )(g)(f x and )(f)(g x
Solution
xxxxx 8372889)72(49)(gf)(g)(f
xxxxx 81178187)49(2)(fg)(f)(g
Notice that )(g)(f x and )(f)(g x produces different answers )(f)(g)(g)(f xx
Example 2 If 5-3x)(h x and 7x-2x)(g 2x find )(g)(h x and )(h)(g x
Solution
52165723g(x)h)(g)(h 22 xxxxx
8581183521506018
3521)25309(2)53(7)53(2h(x)g)()(g
22
22
xxxxx
xxxxxxh
Exercise
1) For the given functions find the indicated composite function
a) If 24x-x)(f 2 x and 7-3x)(g x , find )(g)(f x and )(f)(g x
b) If 6x-5)(g x and 11-9x)(h x , find )(g)(h x and )(h)(g x
c) If 52)(f xx and 3-5x)(g 2x , find )(g)(f x and )(f)(g x
d) If xx 29)(f and 35xx4)(g 2 x , find )(g)(f x and )(f)(g x
e) If 3)(f xx and 93x-x4)(g 2 x , find )(g)(f x and )(f)(g x
f) If 3 4-x)(f x and 4x)(g 3 x , find )(g)(f x and )(f)(g x
2) Let 42 72)(g and6)(f xxxxx Perform the indicated operation and state the domain
a) )(g)(f xx b) )(g)(f xx c) )(g)(f xx d) )(g)(f xx
3) Let 8)(g and3)(f 1 xxxx Perform the indicated operation and state the domain.a)
)(gf x b) )(fg x c) )(ff x d) )(gg x
Inverse Functions In the previous section you learned that a relation is a mapping of input values onto output
values. An inverse relation maps the output values back to their original input values. This
means that the domain of the inverse relation is the range of the original relation and that the
range of the inverse relation is the domain of the original relation.
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The graph of an inverse relation is the reflection of the graph of the original relation on y = x.
To find the inverse of a relation that is given by an equation in x and y, switch the roles of x and y
and solve for y (if possible).
Example 1 Find an equation for the inverse of the relation 42 xy .
Solution
The original relation is 42 xy . Switch x and y to get 42 yx then solve for y. 22
1 xy
The inverse relation is 22
1 xy
In Example 1 both the original relation and the inverse relation happen to be functions. In such
cases the two functions are called inverse functions.
Remarks: Functions ƒ and g are inverses of each other provided xx )(gf and xx )(fg
The function g is denoted by -1f , read as “ƒ inverse.”
Given any function, you can always find its inverse relation by switching x and y. For a linear
function f(x) = mx + b where m ≠ 0, the inverse is itself a linear function.
Example 2 Verify that 42)(f xx and 2)(f2
1-1 xx are inverses
Solution
We need to show that xx )(ff -1 and xx )(ff -1 Now
xxxxx 222)42()42(f)(ff2
1-1-1 and
xxxxx 444222f)(ff2
1
2
1-1 )(f x and )(f -1 x are inverses
Example 3 When calibrating a spring
scale, you need to know how far the spring
stretches based on given weights. Hooke’s
law states that the length a spring stretches is
proportional to the weight attached to the
spring. A model for one scale is
30.5wl where l is the total length (in
inches) of the spring and w is the weight (in
pounds) of the object.
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a) Find the inverse model for the scale.
b) If you place a melon on the scale and the spring stretches to a total length of 5.5”, how much
does the melon weigh?
Solution
a) 62l0.53)-l(w30.5wl
b) To find the weight of the melon, substitute 5.5 for l to get 562(5.5)62lw
Thus the melon weighs 5 pounds
The graphs of the power functions 2)(f xx
and 3)(g xx are shown below along with
their reflections in the line y = x. Notice that
the inverse of 3)(g xx is a function, but
that the inverse of 2)(f xx is not a
function. But if the domain of 2)(f xx is
restricted, to only nonnegative real numbers,
then the inverse of f is a function.
Example 4 Find the inverse of the function 2)(f xx , x ≥ 0.
Solution 2)(f xxy Switch x and y to get xyyx 2
. Because the domain of ƒ is restricted to
nonnegative values, the inverse function is xx )(f -1
(You would choose xx )(f -1if the
domain had been restricted to x ≤ 0.)
Check To check your work, graph ƒ and -1f
as shown. Note that the graph of
xx )(f -1is the reflection of the graph of
2)(f xx , x ≥ 0 in the line y = x.
In the graphs at the top of the page, notice that the graph of 2)(f xx can be intersected twice
with a horizontal line and that its inverse is not a function. On the other hand, the graph of 3)(g xx cannot be intersected twice with a horizontal line and its inverse is a function. This
observation suggests the horizontal line test.
Horizontal Line Test: If no horizontal line intersects the graph of a function ƒ more than once,
then the inverse of ƒ is itself a function.
Example 5 Consider the function 2)(f 3
2
1 xx . Determine whether the inverse of ƒ is a
function. Then find the inverse.
Solution Begin by graphing the function and noticing that no horizontal line intersects the graph
more than once. This tells you that the inverse of ƒ is itself a function. To find an equation for -1f , complete the following steps.
Write original function as 2y 3
2
1 x ƒ(x) is Replaced with y. Now Switch x and y.
33
2
1 422x xyy Thus the inverse function is 3-1 42)(f xx
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Example 6 Near the end of a star’s life the
star will eject gas, forming a planetary
nebula. The Ring Nebula is an example of a
planetary nebula. The volume V (in cubic
kilometers) of this nebula can be modeled
by 326)1001.9( tV where t is the age (in
years) of the nebula. Write the inverse
model that gives the age of the nebula as a
function of its volume. The graph of the function in Eg5
Then determine the approximate age of the Ring Nebula given that its volume is about 1 38105.1 cubic kilometers.
Solution
393
26
326 )1004.1()1001.9(
)1001.9( VV
ttV
. Substitute 138105.1 for V to get
5500105.1)1004.1()1004.1(3 38939 Vt . The Ring Nebula is about 5500 years old.
Exercise
1) Explain the steps in finding an equation for an inverse function.
2) Find the inverse relation.
X 1 2 3 4 5
Y -1 -2 -3 -4 -5
X -2 -1 0 2 4
y 2 1 0 1 2
X 1 4 1 0 1
y 3 -1 6 -3 9
X 1 -2 4 2 -2
y 0 3 -2 2 -1
3) Find an equation for the inverse relation.
a) x5y b) 12y x c) x3
26y d) 4
13y x e) 138y x f) x7
3
7
5y
4) Verify that ƒ and g are inverse functions a) 22)(g,1)(f2
1 xxxx b)
2
1
6
1)(g,36)(f xxxx c) 21
2
13 )(g,8)(f xxxx d) xxxxx 3)(g,0)(f 2
3
1 e)
31
)1()(g,13)(f2
13 xxxx f) 55
7
1 27)(g,1)2()(f xxxx g)
4
4
14 7)(g,01256)(f xxxxx
5) Find the inverse function; a) 2)(f 3 xx b) 9)(f 3
5
3 xx c) 0,3)(f 4 xxx
d) 5
3
1 2)(f xx e) 0,2)(f 2 xxx
f) 5
6
1
3
2)(f xx g) 0,)(f2
14 xxx
6) The graph of xx 1)(f is shown. Is
the inverse of ƒ a function? Explain
7) Find the inverse power function a) 7)(f xx b) 0,)(f2
16 xxx c) 0,3)(f 4 xxx
d) 5
32
1)(f xx e) 310)(f xx f) 0,)(f 2
4
9 xxx
8) Match the graph with the graph of its inverse
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a) b) c)
9) Graph the function ƒ. Then use the graph to determine whether the inverse of ƒ is a function
a) xx 23)(f b) 3)(f xx c) 1)(f 2 xx d) 22)(f xx e) 3)(f 3 xx
f) 2)(f xx g) 32)(f xx h) )3)(1()(f xxx j) 196)(f 4 xxx
10) The Federal Reserve Bank of New York reports international exchange rates at 12:00 noon
each day. On January 20, 1999, the exchange rate for Canada was 1.5226. Therefore, the
formula that gives Canadian dollars in terms of United States dollars on that day is
USC DD 5226.1 where CD represents Canadian dollars and
USD represents United States
dollars. Find the inverse of the function to determine the value of a United States dollar in
terms of Canadian dollars on January 20, 1999.
11) The formula to convert temperatures from degrees Fahrenheit to degrees Celsius is:
)32(9
5 FC Write the inverse of the function, which converts temperatures from degrees
Celsius to degrees Fahrenheit. Then find the Fahrenheit temperatures that are equal t-29°C,
10°C, and 0°C.
12) In bowling a handicap is a change in score to adjust for differences in players’ abilities. You
belong to a bowling league in which each bowler’s handicap h is determined by his or her
average a using this formula: )200(9.0 ah (If the bowler’s average is over 200, the
handicap is 0.) Find the inverse of the function. Then find your average if your handicap is
27.
13) You and a friend are playing a number-guessing game. You ask your friend to think of a
positive number, square the number, multiply the result by 2, and then add 3. If your friend’s
final answer is 53, what was the original number chosen? Use an inverse function in your
solution.
14) The weight w (in kilograms) of a hake, a type of fish, is related to its length l (in centimeters)
by this function: )1037.9( 6
3
1 w Find the inverse of the function. Then determine the
approximate length of a hake that weighs 0.679 kilogram.
15) The weight w (in pounds) that can be supported by a shelf made from half-inch Douglas fir
plywood can be modeled by
39.82
dw where d is the distance (in inches) between the
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supports for the shelf. Find the inverse of the function. Then find the distance between the
supports of a shelf that can hold a set of encyclopedias weighing 66 pounds.
Graphing Quadratic Function
A quadratic function has the form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 where𝑎 ≠ 0. The graph of a quadratic
function is U-shaped and is called a parabola. Eg the graphs of 𝑦 = 𝑥2 and 𝑦 = −𝑥2 are shown.
The origin is the lowest point on the graph
of 2xy and the highest point on the graph
of 2xy . The lowest or highest point on
the graph of a quadratic function is called
the vertex. The graphs of 2xy and 𝑦 =
−𝑥2are symmetric about the y-axis, called
the axis of symmetry. In general, the axis of
symmetry for the graph of a quadratic
function is a vertical line through the vertex.
Activity: Graph each of these functions on the same axis: of 𝑦 = 0.5𝑥2 , 𝑦 = 𝑥2, 𝑦 =
2𝑥2 and 𝑦 = 3𝑥2 Repeat the exercise for these functions: 𝑦 = −0.5𝑥2 , 𝑦 = −𝑥2, 𝑦 =
−2𝑥2 and 𝑦 = −3𝑥2
i) What are the vertex and axis of symmetry of the graph of , 𝑦 = 𝑎𝑥2?
ii) Describe the effect of a on the graph of , 𝑦 = 𝑎𝑥2
In the above activity, you examined the graph of the simple quadratic function 𝑦 = 𝑎𝑥2. The
graph of the more general function 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is a parabola with these characteristics:
• The parabola opens up if 𝑎 > 0 and opens down if 𝑎 < 0. The parabola is wider than the
graph of 𝑦 = 𝑥2 if |𝑎| < 1 and narrower than the graph of 𝑦 = 𝑥2 if |𝑎| > 1.
• The x-coordinate of the vertex is a
b2
and the axis of symmetry is the vertical line a
bx
2 .
Example 1 Graph 𝑦 = 2𝑥2 − 8𝑥 + 6 . Solution
Note that the coefficients for this function are a = 2, b =-8, and c = 6. Since a > 0, the parabola
opens up. Find and plot the vertex. The x-coordinate is 22
ab .
The y-coordinate is 𝑦 = 2 × 22 − 8 × 2 +
6 = −2 So, the vertex is (2, -2).
Draw the axis of symmetry 𝑥 = 2.
Plot two points on one side of the axis of
symmetry, such as (1, 0) and (0, 6). Use
symmetry to plot two more points, such as
(3, 0) and (4, 6).
Draw a parabola through the plotted points The quadratic function 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is written in standard form Two other useful forms
for quadratic functions are given below.
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Vertex form: 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 The vertex is (h, k). The axis of symmetry is 𝑥 = ℎ.
Intercept form:𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞) The x-intercepts are p and q. The axis of symmetry is
halfway between (p, 0) and (q, 0).
Note For both forms, the graph opens up if 𝑎 > 0 and opens down if 𝑎 < 0.
Example 2 Graph 𝑦 = −1
2(𝑥 + 3)2 + 4
Solution
The function is in vertex form 𝑦 = 𝑎(𝑥 −
ℎ)2 + 𝑘 where 𝑎 = −1
2, ℎ = −3, and 𝑘 =
4. Since 𝑎 < 0, the parabola opens down.
To graph the function, first plot the vertex
(h, k) = (-3, 4). Draw the axis of symmetry x
= -3 and plot two points on one side of it,
such as (-1, 2) and (1, -4). Use symmetry to
complete the graph
Example 3 Graph 𝑦 = −(𝑥 + 2)(𝑥 − 4)
Solution
The function is in intercept form 𝑦 = 𝑎(𝑥 −𝑝)(𝑥 − 𝑞)where 𝑎 = −1, 𝑝 = −2, and 𝑞 =4. the x-intercepts occur at (-2, 0) and (4, 0).
The axis of symmetry lies halfway between
these points, at 𝑥 = 1. So, the x-coordinate
of the vertex is 𝑥 = 1 and the y-coordinate
of the vertex is: 𝑦 = −(1 + 2)(1 − 4) = 9
The graph of the function is shown.
Example 4 Researchers conducted an experiment to determine temperatures at which people
feel comfortable. The percent y of test subjects who felt comfortable at temperature x (in degrees
Fahrenheit) can be modeled by: 𝑦 = −3.678𝑥2 + 527.3𝑥 − 18,807
a) What temperature made the greatest percent of test subjects comfortable?
b) At that temperature, what percent felt comfortable?
Solution
a) Since a is negative, the graph opens down and the function has a maximum value.
The maximum value occurs at: 72)678.3(2
3.527
2
a
bx
b) The corresponding value of y is: : 𝑦 = −3.678(72)2 + 527.3(72) − 18,807 ≈ 92 The
temperature that made the greatest percent of test subjects comfortable was about 72°F. At
that temperature about 92% of the subjects felt comfortable.
Example 5 The Golden Gate Bridge in San Francisco has two towers that rise 500 feet above
the road and are connected by suspension cables as shown. Each cable forms a parabola with
equation 𝑦 =1
8960(𝑥 − 2100)2 + 8 where x and y are measured in feet. Source: Golden Gate
Bridge, Highway and Transportation District
a) What is the distance d between the two towers?
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b) What is the height ¬ above the road of a cable at its lowest point? S
Solution
a) The vertex of the parabola is (2100, 8), so a cable’s lowest point is 2100 feet from the left
tower shown above. Since the heights of the two towers are the same, the symmetry of the
parabola implies that the vertex is also 2100 feet from the right tower. Therefore, the towers
are 𝑑 = 2(2100) = 4200 feet apart.
b) The height ¬ above the road of a cable at its lowest point is the y-coordinate of the vertex.
Since the vertex is (2100, 8), this height is 𝑙 = 8 feet.
Exercise
1) Graph the quadratic function. Label the vertex and axis of symmetry. 742 xxy
4)1(2 2 xy )1)(2( xxy 322
3
1 xxy 6)4( 2
5
3 xy )3(2
5 xxy
2) Is )8)(5(2 xxy in standard form, vertex form, or intercept form?
3) Write the quadratic function in standard form )1)(2( xxy )3)(4(2 xxy
5)1(4 2 xy 7)2( 2 xy )8)(6(2
1 xxy 4)9( 2
3
2 xy
4) The equation given in Example 5 is based on temperature preferences of both male and
female test subjects. Researchers also analyzed data for males and females separately and
obtained the equations below.
Males: 217736.6124290 2 xxy Females: 330929.9086224 2 xxy
What was the most comfortable temperature for the males? for the females?
5) Match the quadratic function with its graph a) )3)(2( xxy b) 2)3( 2 xy c)
1162 xxy
6) Graph the quadratic function. Label the vertex and axis of symmetry .a) 122 xxy
b) 19122 2 xxy c) 242 xxy d) 53 2 xy e) 542
2
1 xxy
f) 32
6
1 xxy g) 5)4(3 2 xy h) 4)3(2 2 xy i) 3)1( 2
3
1 xy
j) 2
4
5 )3( xy
7) Graph the quadratic function. Label the vertex, axis of symmetry, and x-intercepts.
a) )6)(2( xxy b) )1)(1(4 xxy c) )5)(3( xxy
d) )1)(4(3
1 xxy e) )3)(2(2
1 xxy f) )2(3 xxy
8) Write the quadratic function in standard form.
a) )5)(2( xxy b) )4)(3( xxy c) )7)(4(3 xxy d)
)14)(85( xxy e) 2)3( 2 xy f) 2)5(11 xy g) 9)2(6 2 xy
h) 20)7(8 2 xy i) 2)29(4 xxy j) )3)(6(3
7 xxy k) 2
32
2
1 )18( xy
9) In parts (a) and (b), use a graphing calculator or computer to examine how b and c affect the
graph of cbxaxy 2
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a) Graph cxy 2 c for c = -2, -1, 0, 1, and 2. Use the same viewing window for all the
graphs. How do the graphs change as c increases?
b) Graph y = x2 + bx for b = -2, -1, 0, 1, and 2. Use the same viewing window for all the
graphs. How do the graphs change as b increases?
10) Although a football field appears to be flat, its surface is actually shaped like a parabola so
that rain runs off to either side.
The cross section of a field with
synthetic turf can be modeled by
5.1)80(000234.0 2 xy where x
and y are measured in feet. What is the
field’s width? What is the maximum
height of the field’s surface? 11) Scientists determined that the rate y (in calories per minute) at which you use energy while
walking can be modeled by 15050,3.51)2.90(00849.0 2 xxy where x is your
walking speed (in meters per minute).
a) Graph the function on the given domain.
b) Describe how energy use changes as walking speed increases.
c) What speed minimizes energy use?
12) The woodland jumping mouse can hop
surprisingly long distances given its
small size. A relatively long hop can be
modeled by )6(9
2 xxy where x and
y are measured in feet. How far can a
woodland jumping mouse hop? How
high can it hop?
13) The engine torque y (in foot-pounds) of one model of car is given by
8.382.2375.3 2 xxy where x is the speed of the engine (in thousands of revolutions
per minute). Find the engine speed that maximizes torque. What is the maximum torque?
14) A kernel of popcorn contains water that expands when the kernel is heated, causing it to pop.
The equations below give the “popping volume” y (in cubic centimeters per gram) of
popcorn with moisture content x (as a percent of the popcorn’s weight).
Hot-air popping: 8.944.21761.0 2 xxy Hot-oil popping: 0.767.17652.0 2 xxy
a) For hot-air popping, what moisture content maximizes popping volume? What is the
maximum volume?
b) For hot-oil popping, what moisture content maximizes popping volume? What is the
maximum volume?
c) The moisture content of popcorn typically ranges from 8% to 18%. Graph the equations
for hot-air and hot-oil popping on the interval 8 ≤ x ≤ 18.
d) Based on the graphs from part (c), what general statement can you make about the
volume of popcorn produced from hot-air popping versus hot-oil popping for any
moisture content in the interval 8 ≤ x ≤ 18?
15) Write khxay 2)( and ))(( qxpxay in standard form. Knowing that the vertex
of the graph of cbxaxy 2 occurs at a
bx2
, show that the vertex for khxay 2)(
occurs at x = h and that the vertex for ))(( qxpxay occurs at 2
qpx
.
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Graphing Absolute Value Functions
The absolute value of x is defined by
0for
0for)(f
xx
xxxx
The graph of this piece wise function consists of two rays, is V-shaped, and opens up.
The corner point of the graph, called the vertex, occurs at the origin.
Notice that the graph of y = |x| is symmetric about the y-axis because for every point (x, y) on the
graph, the point (-x, y) is also on the graph
Activity
1) In the same coordinate plane, graph y = a|x| for a -2, , 21
2
1 and 2. What effect does a have
on the graph of y = a|x|? What is the vertex of the graph of y = a|x|?
2) In the same coordinate plane, graph y = |x- h| for h = -2, 0, and 2. What effect does h have on
the graph of y = |x-h|? What is the vertex of the graph of y = |x-h|?
3) In the same coordinate plane, graph y = |x| + k for k = -2, 0, and 2. What effect does k have on
the graph of y = |x| + k? What is the vertex of the graph of y = |x| + k?
Although in the above exercise you investigated the effects of a, h, and k on the graph of
𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 separately, these effects
can be combined. Eg, the graph of y = |x| is
shown in blue along with the graph of.
342 xy in red. Notice that the vertex
of the red graph is (4, 3) and that the red
graph is narrower than the blue graph. The graph of 𝑦 = 𝑎|𝑥 − ℎ| + 𝑘 has the following characteristics.
i) The graph has vertex (h, k) and is symmetric in the line x = h.
ii) The graph is V-shaped. It opens up if a > 0 and down if a < 0.
iii) The graph is wider than the graph of y = |x| if |a| < 1.
iv) The graph is narrower than the graph of y = |x| if |a| > 1.
To graph an absolute value function you may find it helpful to plot the vertex and one other
point. Use symmetry to plot a third point and then complete the graph.
Example 1 Graph 𝑦 = −|𝑥 + 2| + 3.
Solution
To graph 𝑦 = −|𝑥 + 2| + 3, plot the vertex
at (-2, 3). Then plot another point on the
graph, such as (-3, 2). Use symmetry to plot
a third point, (-1, 2). Connect these three
points with a V-shaped graph. Note that 𝑎 =−1 < 0 and |a| = 1, so the graph opens
down and is the same width as the graph of y
=|x|.
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Example 2 Write an equation of the graph shown.
Solution
The vertex of the graph is (0, -3), so the equation has the form: 𝑦 = 𝑎|𝑥 − 0| + (−3) or
3 xay
To find the value of a, substitute the
coordinates of the point (2, 1) into the
equation and solve.
2321 aa . So the equation of the
graph is 32 xy .
Note The graph opens up and is narrower
than the graph of y = |x|, so 2 is a reasonable
value for a.
Example 3 The front of a camping tent can be modeled by the function 5.35.24.1 xy
where x and y are measured in feet and the x-axis represents the ground. a. Graph the function.
b. Interpret the domain and range of the function in the given context.
Solution
a) The graph of the function is shown. The
vertex is (2.5, 3.5) and the graph opens
down. It is narrower than the graph of y
= |x|.
b) The domain is 0 ≤ x ≤ 5, so the tent is 5
feet wide. The range is 0 ≤ y ≤ 3.5, so
the tent is 3.5 feet tall.
Example 4 While playing pool, you try to shoot the eight ball into the corner pocket as shown.
Imagine that a coordinate plane is placed over the pool table.
The eight ball is at ),5(4
5 and the pocket you
are aiming for is at (10, 5). You are going to
bank the ball off the side at (6, 0).
a) Write an equation for the path of the
ball.
b) Do you make your shot?
Solution
a) The vertex of the path of the ball is (6, 0), so the equation has the form 𝑦 = 𝑎|𝑥 − 6|.
Substitute the coordinates of the point ),5(4
5 into the equation and solve for a.
b) 4
5
4
565 aa . So the equation for the path of the ball is 64
5 xy .
a. You will make your shot if the point (10, 5) lies on the path of the ball. 61054
5
Substitute 5 for y and 10 for x. 55 _ Simplify. The point (10, 5) satisfies the equation, so
you do make your shot
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Exercise
1) Graph the function. Then identify the vertex, tell whether the graph opens up or down, and
whether the graph is wider, narrower, or the same width as the graph of y = |x|. a) xy2
1
b) 2
5 xy c) 2
3
3
2 xy d) 1062 xy e) 22
1 xy f) 433
1 xy
2) Write an equation for the function whose graph is shown.
3) Which statement is true about the graph
of the function y =-|x + 2| + 3?
i) Its vertex is at (2, 3).
ii) Its vertex is at (-2, -3).
iii) It opens down.
iv) It is wider than the graph of y = |x|.
4) Suppose that the tent in Example 3 is 7 feet wide and 5 feet tall. Write a function that models
the front of the tent. Let the x-axis represent the ground. Then graph the function and identify
the domain and range of the function.
5) Match the function with its graph. a) xy 3 b) xy 3 c) xy31
6) Use the Trace feature of the graph of absolute functions to find the corresponding x-value(s)
for the given y-value; a) 10;4 yxy b) 9;14 yxy c) 2
3
8
15 ; yxy d)
0;57
4 yxy e)2
1;52 yxy f) 2;732 yxy g)
4
25
2
3 ;63 yxy h) 5;55.175.3 yxy
7) Match the function with its graph. a) 2 xy b) 2 xy c) 2 xy
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8) A musical group’s new single is released. Weekly sales s (in thousands) increase steadily for
a while and then decrease as given by the function 40202 ts where t is the time (in
weeks) Graph the function hence find the maximum number of singles sold in one week
9) A rainstorm begins as a drizzle, builds up to a heavy rain, and then drops back to a drizzle.
The rate r (in inches per hour) at which it rains is given by the function 2
1
2
1 1 ts
where t is the time (in hours).
a) Graph the function.
b) For how long does it rain and when does it rain the hardest?
10) Suppose a musical piece calls for an orchestra to start at fortissimo (about 90 decibels),
decrease in loudness to pianissimo (about 50 decibels) in four measures, and then increase
back to ortissimo in another four measures. The sound level s (in decibels) of the musical
piece can be modeled by the function 5410 ms where m is the number of measures.
a) Graph the function for 0 ≤ m ≤ 8.
b) After how many measures should the orchestra be at the loudness of mezzo forte
(about 70 decibels)?
11) You are sitting in a boat on a lake. You can get a sunburn from sunlight that hits you directly
and from sunlight that reflects off the water. Sunlight reflects off the water at the point (2, 0)
and hits you at the point (3.5, 3). Write and graph the function that shows the path of the
sunlight.
12) The Transamerica Pyramid, shown at the right, is an office building in San Francisco. It
stands 853 feet tall and is 145 feet wide at its base. Imagine that a coordinate plane is placed
over a side of the building. In the coordinate plane, each unit represents one foot
, and the origin is at the center of the
building’s base. Write an absolute
function whose graph is a V-shaped
outline of the sides of the building,
ignoring the “shoulders” of the
building.
13) You are trying to make a hole in-one on the miniature golf green shown. Imagine that a
coordinate plane is placed over the golf green.
The golf ball is at (2.5, 2) and the
hole is at (9.5, 2). You are going to
bank the ball off the side wall of the
green at (6, 8). Write an equation for
the path of the ball and determine if
you make your shot.
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Graphing Simple rational function
A rational function is a function of the form )(
)()(f
xq
xpx where 𝑝(𝑥) and 𝑞(𝑥) are polynomials
and 𝑞(𝑥) ≠ 0. In this lesson you will learn to graph rational functions for which p(x) and q(x) are
linear. For instance, consider the rational function: x
y 1 . The graph of this function is called a
hyperbola and is shown below. Notice the following properties.
• The x-axis is a horizontal asymptote.
• The y-axis is a vertical asymptote.
• The domain and range are all nonzero real numbers.
The graph has two symmetrical parts called branches. For each point (x, y) on one branch, there
is a corresponding point (-x , -y) on the other branch
Activity Graph each function. xxxx
yyyy 2132 ,,,
a) Use the graphs to describe how the sign of a affects the graph of x
ay .
b) Use the graphs to describe how |a| affects the graph of x
ay
All rational functions of the form kyhx
a
have graphs that are hyperbolas with asymptotes at
x = h and y = k. To draw the graph, plot a couple of points on each side of the vertical asymptote.
Then draw the two branches of the hyperbola that approach the asymptotes and pass through the
plotted points..
Example 1 Graph 13
2
xy . State the
domain and range
Solution
Draw the asymptotes x =-3 and y = -1.
Plot two points to the left of the vertical
asymptote, Eg (-4, 1) & (-5, 0), and two
points to the right, such as (-1, -2) and (0,
35 ) Use the asymptotes and plotted points
to draw the branches of the hyperbola. The
domain is all real numbers except -3, and the
range is all real numbers except -1. Note All
rational functions of the form dcx
baxy
also
have graphs that are hyperbolas.
The vertical asymptote occurs at the x-value that makes the denominator zero. The horizontal asymptote is the line cay
Example 2 Graph 42
1
x
xy State the domain and range.
X y
x y
-4 41 4 4
1
-3 31 3 3
1
-2 21 2 2
1
-1 -1 1 1
21 -2 2
1 2
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 77
Solution
Draw the asymptotes. Solve 2𝑥 − 4 = 0 for
x to find the vertical asymptote x = 2. The
horizontal asymptote is the line 21
cay _.
Plot two points to the left of the vertical
asymptote, such as (0, 41 ) and (1, -1), and
2 points to the right, such as (3, 2) & (4, 45 )
Use the asymptotes and plotted points to
draw the branches of the hyperbola.
The domain is all real numbers except 2, and
the range is all real numbers except 21
Example 3 For a fundraising project, your math club is publishing a fractal art calendar. The
cost of the digital images and the permission to use them is $850. In addition to these “one-time”
charges, the unit cost of printing each calendar is $3.25.
a) Write a model that gives the average cost per calendar as a function of the number of
calendars printed.
b) Graph the model and use the graph to estimate the number of calendars you need to print
before the average cost drops to $5 per calendar.
c) Describe what happens to the average cost as the number of calendars printed increases.
Solution
a) The average cost (A) is the total cost of making the calendars divided by the number of
calendars printed
Let x be number of calendars printed then, One-time charges =$850 and the
Unit cost=$3.25 per calendar x
xA
3.25 850
b) The graph of the model is shown at the right. The A-axis is the vertical asymptote and the
line A = 3.25 is the horizontal asymptote. The domain is 𝑥 > 0 and the range is 𝐴 > 3.25When A=5, x is about 500. So, you need
to print about 500 calendars before the
average cost drops to $5 per calendar.
c) As the number of calendars printed
increases, the average cost per calendar
gets closer and closer to $3.25. For
instance, when x = 5000 the average cost
is $3.42, and when x = 10,000 the
average cost is $3.34.
Exercise 1) If the graph of a rational function is a hyperbola with the x-axis and the y-axis as asymptotes,
what is the domain of the function? What is the range?
2) Explain why the graph shown below is not the graph of 73
6 x
y
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 78
3) Identify the horizontal and vertical
asymptotes of the graph of the function
a) 43
2
xy b)
4
32
x
xy
c) 3
3
x
xy d)
42
5
x
xy
e) 108
3
xy f) 5
6
4
xy
4) Look back at Example 3 on page 542. Suppose you decide to generate your own fractals on a
computer to save money. The cost for the software (a “one-time” cost) is $125. Write a
model that gives the average cost per calendar as a function of the number of calendars
printed. Graph the model and use the graph to estimate the number of calendars you need to
print before the average cost drops to $5 per calendar.
5) Identify the horizontal and vertical asymptotes of the graph of the function. Then state the
domain and range a) 23
xy b) 2
3
4
xy c) 2
3
2
xy d)
3
2
x
xy e)
13
22
x
xy f)
54
23
x
xy g) 17
43
22
xy h)
416
234
x
xy i) 19
6
4
xy
6) Match the function with its graph. a) 32
3
xy b) 3
2
3
xy c)
3
2
x
xy
7) Graph the function. State the domain and range a) x
y4
b) 13
3
xy c) 8
5
4
xy d)
37
1
xy e) 6
2
6
xy f)
4
5
xy g) 2
124
1
xy h)
xy
2
3 i) 5
63
4
xy
8) Graph the function. State the domain and range. a) 3
2
x
xy b)
34
x
xy c)
83
7
x
xy
d) 23
19
x
xy e)
124
310
x
xy f)
x
xy
4
25 g)
42
3
x
xy h)
15
7
x
xy i)
12
414
x
xy
9) Write a rational function that has the vertical asymptote 4x and the horizontal asymptote 3y .
10) You’ve paid $120 for a membership to a racquetball club. Court time is $5 per hour.
a) Write a model that represents your average cost per hour of court time as a function of
the number of hours played. Graph the model. What is an equation of the horizontal
asymptote and what does the asymptote represent?
b) Suppose that you can play racquetball at the YMCA for $9 per hour without being a
member. How many hours would you have to play at the racquetball club before your
average cost per hour of court time is less than $9?
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 79
11) Air temperature affects how long it takes sound to travel a given distance. The time it takes
for sound to travel one kilometer can be modeled by 3316.0
1000
Tt where t is the time (in
seconds) and T is the temperature (in degrees Celsius). You are 1 kilometer from a lightning
strike and it takes you exactly 3 seconds to hear the sound of thunder. Use a graph to find the
approximate air temperature. (Hint: Use tick marks that are 0.1 unit apart on the t-axis.)
12) Economist Arthur Laffer argues that beyond a certain percent pm, increased taxes will
produce less government revenue. His theory is illustrated in the graph below.
a) Using Laffer’s theory, an economist
models the revenue generated by one
kind of tax by 110
800080
p
pR
where R is the government revenue
(in tens of millions of dollars) and p
is the percent tax rate (55 ≤ p ≤ 100).
Graph the model.
b) Use your graph from part a to find
the tax rate that yields $600 million
of revenue. 13) When the source of a sound is moving relative to a stationary listener, the frequency ƒl (in
hertz) heard by the listener is different from the frequency ƒs (in hertz) of the sound at its
source. An equation for the frequency heard by the listener is r
740
f740f s
lwhere r is the
speed (in miles per hour) of the sound source relative to the listener.
a) The sound of an ambulance siren has a frequency of about 2000 hertz. You are standing
on the sidewalk as an ambulance approaches with its siren on. Write the frequency that
you hear as a function of the ambulance’s speed.
b) Graph the function from Exercise 47 for 0 ≤ r ≤ 60. What happens to the frequency you
hear as the value of r increases?
14) In what line(s) is the graph of xy 1 asymmetric? What does this symmetry tell you about the
inverse of the function xx 1)(f ?
15) Show algebraically that the function 105
3)(f
xx and the function
5
4710)(g
x
xx are
equivalent.
Graphing General Rational Functions In this section you will learn how to graph rational functions for which p(x) and q(x) may be
higher-degree Let p(x) and q(x) be polynomials with no common factors other than 1.
The graph of the rational function 01
2
2
1
1
01
2
2
1
1
...
...
)(
)()(f
bxbxbxbxb
axaxaxaxa
xq
xpx
n
n
n
n
n
n
m
nm
has the
following characteristics.
i) The x-intercepts of the graph of ƒ are the real zeros of p(x).
ii) The graph of ƒ has a vertical asymptote at each real zero of q(x).
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 80
iii) The graph of ƒ has at most one horizontal asymptote if;
• m < n, the line y = 0 is a horizontal asymptote.
• m = n, the line n
m
b
ay is a horizontal asymptote.
• m > n, the graph has no horizontal asymptote The graph’s end behavior is the same as
the graph of nm
b
axy
n
m
Example 1 Graph 1
42
x
y . State the domain and range.
Solution
The numerator has no zeros, so there is no x-
intercept. The denominator has no real
zeros, so there is no vertical asymptote. The
degree of the numerator is < the degree of
the denominator, so the line y = 0 is a
horizontal asymptote. The bell-shaped graph
passes through the points (-3, 0.4), (-1, 2),
(0, 4), (1, 2), and (3, 0.4). The domain is all
real numbers, and the range is 0 < y ≤ 4
Example 2 Graph 4
32
2
x
xy ..
Solution
The numerator has 0 as its only zero, so the graph has one x-intercept at (0, 0). The denominator
can be factored as (x + 2)(x-2), so the denominator has zeros -2 and 2. This implies that the lines
x = -2 and x = 2 are vertical asymptotes of the graph.
The degree of the numerator (2) is equal to the degree of the denominator (2), so the horizontal
asymptote is 3n
m
b
ay . To draw the graph, plot points between and beyond the vertical
asymptotes.
Example 3 Graph 4
322
x
xxy ..
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 81
Solution
The numerator can be factored as (x-3)(x + 1), so the x-intercepts of the graph are 3 and -1. The
only zero of the denominator is -4, so the only vertical asymptote is x = -4. The degree of the
numerator (2) is greater than the degree of the denominator (1), so there is no horizontal
asymptote and the end behavior of the graph of ƒ is the same as the end behavior of the graph of
xxy 12. To draw the graph, plot points to the left and right of the vertical asymptote.
Manufacturers often want to package their products in a way that uses the least amount of
packaging material. Finding the most efficient packaging sometimes involves finding a local
minimum of a rational function.
Example 4 A standard beverage can has a volume of 355 cubic centimeters.
a) Find the dimensions of the can that has this volume and uses the least amount of material
possible.
b) Compare your result with the dimensions of an actual beverage can, which has a radius of 3.1
centimeters and a height of 11.8 centimeters.
Solution
a) The volume must be 355 cubic centimeters, so you can write the height h of each possible
can in terms of its radius r.
2
2 355355
rhhrv
Using the least amount of material is equivalent to having a minimum surface area S. You
can find the minimum surface area by writing its formula in terms of a single variable and
graphing the result.
rr
rrrhrrS
7102
35522)(2 2
2
2
Graph the function for the surface area S.
Using a graphing calculator, the
Minimum feature gives the minimum
value of S about 278, which occurs when
r ≈ 3.84 centimeters and
66.7)84.3(
3552
h cm
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 82
b) An actual beverage can is taller and narrower than the can with minimal surface area—
probably to make it easier to hold the can in one hand.
Exercise
1) Let )(
)()(f
xq
xpx where p(x) and q(x) are polynomials with no common factors other than 1.
Describe how to find the x-intercepts and the vertical asymptotes of the graph of ƒ.
2) Let )(
)()(f
xq
xpx where p(x) and q(x) are both cubic polynomials with no common factors
other than 1. The leading coefficient of p(x) is 8 and the leading coefficient of q(x) is 2.
Describe the end behavior of the graph of ƒ.
3) Graph the function.
a) 3
62
x
y b)1
42
x
xy c)
2
72
2
x
xy d)
72
3
x
xy e)
1
22
2
x
xy f)
162
x
xy
4) The can for a popular brand of soup has a volume of about 342 cubic centimeters. Find the
dimensions of the can with this volume that uses the least metal possible. Compare these
dimensions with the dimensions of the actual can, which has a radius of 3.3 centimeters and a
height of 10 centimeters
5) Identify the x-intercepts and vertical asymptotes of the graph of the function.
a) 92
x
xy b)
1
32 2
x
xy c)
16
5922
2
x
xxy d)
3
32
x
xy
e)
6
542
x
xxy
f) 8
101332
2
x
xxy g)
92
822
x
xy h)
1
22
2
x
xxy i)
x
xy
2
273
6) Match the function with its graph a) 2
72
2
x
xy b)
4
82
xy c)
42
3
x
xy
7) Give an example of a rational function whose graph has two vertical asymptotes: x=2 and x=7.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 83
8) Graph the function. a) 2
32 2
x
xy b)
8
242
xy c)
1
142
x
xy d)
45
1322
2
xx
xxy
e) 63
2 2
x
xy f)
324
133
3
x
xy g)
27
12113
2
x
xxy h)
145
42
xx
xy
i) 45
1322
2
xx
xxy j)
16
42
2
x
xy k)
x
xxy
2
2092 l)
xx
xxy
4
152
23
9) Match the function with its graph. a) 27
33
x
y b) 92
3
x
xy c)
12
42
x
xxy
10) Suppose you want to make a rectangular garden with an area of 200 square feet. You want to
use the side of your house for one side of the garden and use fencing for the other three sides.
Find the dimensions of the garden that minimize the length of fencing needed.
11) The total energy expenditure E (in joules per gram mass per kilometer) of a typical
budgerigar parakeet can be modeled by v
vvE
75.4717.2131.0 2 where v is the speed of
the bird (in kilometers per hour). Graph the model. What speed minimizes a budgerigar’s
energy expenditure?
12) The mean temperature T (in degrees Celsius) of the Atlantic Ocean between latitudes 40°N
and 40°S can be modeled by 10007403
20000178002
dd
dT where d is the depth (in meters). Graph
the model. Use your graph to estimate the depth at which the mean temperature is 4°C.
13) For 1985 to 1995, the average daily cost per patient C (in dollars) at community hospitals in
the United States can be modeled by 10001225
462048204072
xx
xC where x is the number of years
since 1985. Graph the model. Would you use this model to predict patient costs in 2005?
14) For 1980 to 1995, the total revenue R (in billions of dollars) from parking and automotive
service and repair in the United States can be modeled by 1000268257..0
30432641642723
2
xxx
xxR
where x is the number of years since 1980. Graph the model. In what year was the total
revenue approximately $75 billion?
15) The acceleration due to gravity g§ (in meters per second squared) of a falling object at the
moment it is dropped is given by the function 1372
14
1007.4)1028.1(
1099.3'
hhg where h is
the object’s altitude (in meters) above sea level.
a) Graph the function.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 84
b) What is the acceleration due to gravity for an object dropped at an altitude of
c) 5000 kilometers?
d) Describe what happens to g’ as h increases.
16) For 1980 to 1995, the total revenue R (in billions of dollars) from hotels and motels in the
United States can be modeled by x
xR
01.01
88.2676.2
where x is the number of years since
1980. Graph the model. Use your graph to find the year in which the total revenue from
hotels and motels was approximately $68 billion.
17) Consider the following two functions )5)(3(
)3)(2()(g and
)5)(3(
)2)(1()(f
xx
xxx
xx
xxx Notice
that the numerator and denominator of g have a common factor of x- 3.
a) Make a table of values for each function from x = 2.95 to x = 3.05 in increments of 0.01.
b) Use your table of values to graph each function for 2.95 ≤ x ≤ 3.05.
c) As x approaches 3, what happens to the graph of ƒ(x)? to the graph of g(x)?
Graphing Exponential and Power Functions
An Exponential Functions involves the expression 𝑏𝑥 where b is a positive number other than 1
To see the basic shape of the graph of an exponential function such as ƒ(𝑥) = 2𝑥 , you can make
a table of values and plot points, as shown
below
x -3 -2 -1 0 1 2 3
𝑦 = 2𝑥 0.125 0.25 0.5 1 2 4 8
Notice the end behavior of the graph. As
x )(f x , which means that the
graph moves up to the right. As
0)(f, xx which means that the
graph has the line y = 0 as an asymptote. An
asymptote is a line that a graph approaches
as you move away from the origin.
Activity Graph y = 0.2(2)𝑥 and y = 5(2)𝑥. Compare the graphs with the graph of y = 2𝑥.
Also Graph y = −0.2(2)𝑥and y = −5(2)𝑥 and compare the graphs with the graph of y = 2𝑥.
Describe the effect of a on the graph of y = 𝑎(2)𝑥 when a is positive and when a is negative
In the activity you may have observed the following about the graph of y = 2𝑥: The graph
passes through the point (0, a). That is, the y-intercept is a and the x-axis is an asymptote of the
graph.. The domain is all real numbers and the range is y > 0 if a > 0 and y < 0 if a < 0. These
characteristics of the graph of y = 2𝑥 applies for any graph of the form 𝑦 = 𝑎𝑏𝑥 . If 𝑎 >
0 and 𝑏 > 1, the function 𝑦 = 𝑎𝑏𝑥 is an exponential growth function
Example Graph the function; a) y = 0.5(3)𝑥 b) y = −1.5𝑥
Solution
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 85
Plot (0,0.5) and (1,1.5) . Then, from left to
right, draw a curve that begins just above the
x-axis, passes through the two points, and
moves up to the right. .
Plot (0, −1) and (1, −1.5) then, from left to
right, draw a curve that begins just below
the x-axis, passes through the two points,
and moves down to the right.
To graph a general exponential function, kaby hx , begin by sketching the graph of = 𝑎𝑏𝑥 .
Then translate the graph horizontally by h units and vertically by k units.
Example Graph 423 1 xy State the
domain and range.
Solution
Begin by lightly sketching the graph xy 23 , which passes through (0, 3) and
(1, 6). Then translate the graph 1 unit to the
right and 4 units down. Notice that the graph
passes through (1, -1) and (2, 2). The
graph’s asymptote is the line 4y . The
domain is all real numbers, and the range is
4y .
Modelling with an Exponential Function
Just as two points determine a line, two points also determine an exponential curve.
Example Write an exponential function 𝑦 = 𝑎𝑏𝑥 whose graph passes through (1, 6) & (3, 24).
Solution
Substitute the two given points into 𝑦 = 𝑎𝑏𝑥 to obtain two equations in 𝑎 and 𝑏.
Thus 6 = 𝑎𝑏1 and 24 = 𝑎𝑏3. To solve the system, divide equation iI by equation I to get 𝑏2 =
4 , 0r 𝑏 = ±. Using 𝑏 = 2, you then have 𝑎 = 3. So, = 3 (2)𝑥 . .
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 86
When you are given more than two points, you can decide whether an exponential model fits the
points by plotting the natural logarithms of the y-values against the x-values. If the new points
(x, ln y) fit a linear pattern, then the original points (x, y) fit an exponential pattern.
Example 2 The table gives the number y (in millions) of cell-phone subscribers from 1988 to
1997 where t is the number of years since 1987.
t 1 2 3 4 5 6 7 8 9 10
y 1.6 2.7 4.4 6.4 8.9 13.1 19.3 28.2 38.2 48.7
a) Draw a scatter plot of ln y versus x. Is an exponential model a good fit for the original data?
b) Find an exponential model for the original data.
Solution
Use a calculator to create a new table of values.
t 1 2 3 4 5 6 7 8 9 10
𝑙𝑛 𝑦 0.47 0.99 1.48 1.86 2.19 2.57 2.96 3.34 3.64 3.89
Then plot the new points as shown. The points lie close to a line, so an exponential model should
be a good fit for the original data.
To find an exponential model 𝑦 = 𝑎𝑏𝑥 ,
choose two points on the line, such as (2,
0.99) and (9, 3.64). Use these points to find
an equation of the line. Then solve for y.
teeey
y
)46.1(30.1
0.2339t37.0ln
9t37.00.2330.2339t37.0
A graphing calculator that performs
exponential regression does essentially what
is done in Example 2, but uses all of the
original data.
Power function
A power function has the form 𝑦 = 𝑎𝑥𝑏 . Because there are only two constants (a and b), only
two points are needed to determine a power curve through the points.
Example 3 Write a power function 𝑦 = 𝑎𝑥𝑏 whose graph passes through (2, 5) and (6, 9).
Solution
Substitute the coordinates of the 2 given points into 𝑦 = 𝑎𝑥𝑏 to obtain two equations in a and b
5 = 𝑎2𝑏 and 9 = 𝑎6𝑏 To solve the system, solve for a in the first equation to get 𝑎 = 5(2)−𝑏,
then substitute into the second equation 0.5353log
8.1log8.1396)2(5 bbbb
Using b = 0.535, you then have𝑎 = 5(2)−0.535 ≈ 3.45., so 0.5353.45xy
When you are given more than two points, you can decide whether a power model fits the points
by plotting the natural logarithms of the y-values against the natural logarithms of the x-values.
If the new points (ln x, ln y) fit a linear pattern, then the original points (x, y) fit a power pattern.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 87
Example 4 The table gives the mean distance x from the sun (in astronomical units) and the
period y (in Earth years) of the six planets closest to the sun.
Planet Mercury Venus Earth Mass Jupitor Sartun
X 0.387 0.723 1.000 1.524 5.203 9.539
Y 0.214 0.615 1.000 1.881 11.852 29.458
a) Draw a scatter plot of ln y versus ln x. Is a power model a good fit for the original data?
b) Find a power model for the original data.
c) Use the model to estimate the period of Neptune, which has a mean distance from the sun
of 30.043 astronomical units.
Solution
a) Use a calculator to create a new table of values.
𝑙𝑛 𝑥 -0.949 -0.324 0.000 0.421 1.629 2.255
𝑛 𝑦 -1.423 -0.486 0.000 0.632 2.473 3.383
Then plot the new points, as shown at the
right. The points lie close to a line, so a
power model should be a good fit for the
original data.
b) To find a power model y = axb , choose
two points on the line, such as (0, 0) and
(2.255 , 3.383). Use these points to find
an equation of the line. Then solve for y.5.1ln5.1ln xyxy
c) Substituting 30.043 for x in the model
gives 𝑦 = (30.043)1.5 ≈ 165 years for
the period of Neptune
Exercise
1) Write an exponential function of the form xaby whose graph passes through the given
points. a) (1, 3), (2, 36) b) (2, 18), (3, 108) c) (1, 4), (3, 16) d) (2, 3.5), (1, 5.2)
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 88
e) (-3 , 3), (4, 6561) f) ),1(,),4(2
21
81
112 g)(3 , 13.5) , (5 , 30.375) h) ),4(,),2(4
625
4
25
2) Write a power function of the form baxy whose graph passes through; a) (3, 27), (9, 243)
b). (2, 6), (4, 48) c) ),2(,),4(2
1
5
3 d). (4.5, 9.2), (1, 6.4) e) (2.2 , 10.4), (8.8, 20.3) f)
(2.9 , 9.4) , (7.3 , 12.8) g)(2.71 , 6.42) , (13.55 , 29.79)
3) Using xaby and
baxy , take the natural logarithm of both sides of each equation. What is
the slope and y-intercept of the line relating x and ln y for xaby ? of the line relating ln x
and ln y for baxy ?
4) Use the table of values to draw a scatter plot of ln y versus x. Then find an exponential model
for the data.
X 1 2 3 4 5 6 7 8
Y 14 28 56 112 224 448 896 1792
X 1 2 3 4 5 6 7 8
Y 10.2 30.5 43.4 64.2 89.7 120.6 210.4 302.5
X 2 4 6 8 10 12 14 16
Y 12.8 20.48 32.77 52.43 83.89 134.22 214.75 343.6
5) Use the table of values to draw a scatter plot of ln y versus ln x. Then find a power model for
the data..
X 1 2 3 4 5 6 7
Y 0.78 7.37 27.41 69.63 143.47 259.00 426.79
X 1 2 3 4 5 6 7
Y 1.2 5.4 9.8 14.3 25.6 41.2 65.8
X 2 4 6 8 10 12 14
Y 1.89 1.44 1.22 1.09 1.00 0.93 0.87
6) Write y as a function of x 5.424.0log xy b) 8.02.0log xy c) xy 88.012.0log
d) 4ln xy e) 548.0log48.0log xy f) 7.4ln3.2ln xy g)
98.038.2ln xy h) xy log751.348.1log i) xy 5.15.2ln j)
xy log4.3log2.1 k) xy loglog6
5
2
1 l) 8
3
4
1
8
1 ln4ln2 xy
7) Find equations of the line, the exponential curve, and the power curve that each pass through
the points (1, 3) and (2, 12). Graph the equations in the same coordinate plane and then
describe what happens when the equations are used as models to predict y-values for x-values
greater than 2.
8) You have just created your own Web site. You are keeping track of the number of hits (the
number of visits to the site). The table shows the number y of hits in each of the first 10
months where x is the month number
X 1 2 3 4 5 6 7 8 9 10
Y 22 39 70 126 227 408 735 1322 2380 4285
a) Find an exponential model for the data.
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b) According to your model, how many hits do you expect in the twelfth month?
c) According to your model, how many hits would there be in the thirty-fourth month?
What is wrong with this number?
9) The table shows the number C of cranes in Izumi, Japan, from 1950 to 1990 where t
represents the number of years since 1950. Source: Yamashina Institute of Ornithology
T 0 5 10 15 20 25 30 35 40
C 293 299 438 1573 2336 3649 5602 7610 9959
a) Draw a scatter plot of ln C vs t. Is an exponential model a good fit for the original data?
b) Find an exponential model for the original data. Estimate the number of cranes in Izumi,
Japan, in the year 2000.
10) The table shows the cumulative number s of different stamps in the United States from 1889
to 1989 where t represents the number of years since 1889.
T 0 10 20 30 40 50 60 70 80 90 100
S 218 293 374 541 681 858 986 1138 1318 1794 2438
a) Draw a scatter plot of ln s vs t. Is an exponential model a good fit for the original data?
b) Find an exponential model for the original data. Estimate the cumulative number of
stamps in the United States in the year 2000.
11) The table shows the population y (in
millions) and the population rank x for
nine cities in Argentina in 1991.
a) Draw a scatter plot of ln y versus ln
x. Is a power model a good fit for the
original data?
b) Find a power model for the original
data. Estimate the population of the
city Vicente Lopez, which has a
population rank of 20.
City Rank
x Population
(millions), y
Cordoba 2 1.21
Rosario 3 1.12
La Matanza 4 1.11
Mendoza 5 0.77
La Plata 6 0.64
Moron 7 0.64
San Miguel de
Tucuman
8 0.62
Lomas de Zamoras 9 0.57
Mar de Plata 10 0.51
12) The femur is a large bone found in the leg or hind limb of an animal. Scientists use the
circumference of an animal’s femur to estimate the animal’s weight
The table at the right shows the femur
circumference C (in millimeters) and the
weight W (in kilograms) of several
animals.
a) Draw two scatter plots, one of ln W
versus C and another of ln W versus
ln C.
b) Looking at your scatter plots, tell
which type of model you think is a
better fit for the original data.
Explain your reasoning.
Animal C (mm) W (kg)
Meadow mouse 5.5 0.047
Guinea pig 15 0.385
Otter 28 9.68
Cheetah 68.7 38
Warthog 72 90.5
Nyala 97 134.5
Grizzly bear 106.5 256
Kudu 135 301
Giraffe 173 710
c) Using your answer from part (b), find a model for the original data.
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d) The table at the right shows the
femur circumference C (in
millimeters) of four animals. Use the
model you found in part (c) to
estimate the weight of each animal.
Animal C (mm)
Raccoon 28
Cougar 60.25
Bison 167.5
Hippopotamus 208
13) The table shows the atomic number x and the melting point y (in degrees Celsius) for the
alkali metals.
Alkali metal Lithium Sodium Potassium Rubidium Cesium
Atomic number, x 3 11 19 37 55
Melting point, y 180.5 97.8 63.7 38.9 28.5
a) Draw a scatter plot of ln y versus ln x. Is a power model a good fit for the original data?
b) Find a power model for the original data.
c) One of the alkali metals, francium, is not shown in the table. It has an atomic number of
87. Using your model, predict the melting point of francium.
Graphing and Evaluating Polynomial Functions
A polynomial function is a function of the form 01
2
2
1
1 .......)(f axaxaxaxax n
n
n
n
where 0na , the exponents are all whole numbers, and the coefficients are all real numbers. For
this polynomial function, na is the leading coefficient, 0a is the constant term, and n is the
degree. You are already familiar with some types of polynomial functions. Eg, the linear
function f(𝑥) = 3𝑥 + 2 is a polynomial function of degree 1. The quadratic function f(𝑥) =
𝑥2 + 3𝑥 + 2 is a polynomial function of degree 2.
Evaluating Polynomial Functions
One way to evaluate a polynomial function is to use direct substitution. For instance,
7532.)(f 24 xxxx can be evaluated when x = 3 as 987)3(5)3(3)3(2.)3(f 24
Another way of evaluating a polynomial function is to use synthetic substitution. This method
involves fewer operations than direct substitution.
Example 1 The time t seconds it takes a camera battery to recharge after flashing n times can
be modeled by 3.525.00034.0000015.0. 23 nnnt Find the recharge time after 100 flashes.
Solution
Substitute n=100 to get 3.113.5)100(25.0)100(0034.0)100(000015.0. 23 t
The recharge time is about 11 seconds.
Example 2 Use synthetic substitution to evaluate 8452.)(f 24 xxxx when 3x .
Solution
Step 1 Write the coefficients of f(x) in order of decreasing exponent. write the value at which
f(x) is being evaluated to the left as shown below
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Step 2 Bring down the leading coefficient.
Multiply the leading coefficient by the x
value and write the product under the 2nd
coefficient. Then add.
Step 3 Multiply the previous sum by the x
value and write the product under the 3rd
coefficient. Then add. Repeat this procedure
for all the remaining coefficients. The final
sum is the value of f(x) at the given x value.
Synthetic substitution gives f(3)=23 which
agrees with the direct substitution method.
End Behavior of Polynomial Functions
The end behavior of a polynomial’s graph is the behavior of the graph as x approaches positive
infinity or negative infinity. The expression x is read as “x approaches positive infinity.”
The end behavior of a polynomial function is determined by the function’s degree and the sign of
the leading coefficient. The end behavior of the graph of 01
1
1 .......)(f axaxaxax n
n
n
n
:
• For 0na and n even, )(f x as x and )(f x as x
• For 0na and n odd, )(f x as x and )(f x as x
• For 0na and n even, )(f x as x and )(f x as x
• For 0na and n odd, )(f x as x and )(f x as x ‡
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Degree; odd leading coefficient positive Degree; odd leading coefficient negative
Degree; Even leading coefficient positive Degree; Even leading coefficient negative
Example State whether the degree of the polynomial is even or odd. Also state whether the
leading coefficient is positive or negative.
Solution
From the graph )(f x as x so
the degree is even and leading coefficient is
negative.
Graphing Polynomial Functions
To graph a polynomial function, first plot points to determine the shape of the graph’s middle
portion. Then use the knowledge of the end behavior to sketch the ends of the graph.
Example 1 Graph (a) 14.)(f 23 xxxx b) xxxxx 422.)(f 234
Solution
a) To graph the function, make a table of values and plot the corresponding points.
x -3 -2 -1 0 1 2 3
f(𝑥) -7 3 3 -1 -3 3 23
Connect the points with a smooth curve and check the end behavior. The degree is odd and
the leading coefficient is positive, so )(f x as x and )(f x as x . s
b) To graph the function, make a table of values and plot the corresponding points. Connect the
points with a smooth curve and check the end behavior.
x -3 -2 -1 0 1 2 3
f(𝑥) -21 0 -1 0 3 -16 -105
The degree is even and the leading coefficient is negative, so )(f x as x and
)(f x as x
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(a) (b)
Example 2 A rainbow trout can grow up to 40 inches in length. The weight y (in pounds) of a
rainbow trout is related to its length x (in inches) according to the model = 0.0005𝑥3 . Graph the
model. Use your graph to estimate the length of a 10 pound rainbow trout.
Solution
Make a table of values. The model makes sense only for positive values of x.
x 0 5 10 15 20 25 30 35 40
f(𝑥) 0 0.0625 0.5 1.69 4 7.81 13.5 21.4 32
Plot the points and connect them with a
smooth curve, as shown at the right. Notice
that the leading coefficient of the model is
positive and the degree is odd, so the graph
rises to the right. Read the graph backwards
to see that x ≈ 27 when y = 10. A 10 pound
trout is approximately 27 inches long.
Example 3 The energy E in foot-pounds) in each square foot of a wave is modelled b y 𝐸 =0.0029𝑠4 where s is the wind speed (in knots). Graph the model hence estimate the wind speed
need to generate a wave with 1000 foot-pounds of energy per square foot
Solution
Make a table of values. The model only deals with positive values of s
s 0 10 20 30 40
E 0 29 464 2349 7424
Plot the points and join them with a smooth
curve. Because the leading coefficient is
positive and the degree is even the graph
raises upward. From the graph, when
E=1000 the wind speed 24s knots.
Exercise
1) Identify the degree, type, leading coefficient and the constant term for the polynomial
function f(𝑥) = 6 + 2𝑥2 − 5𝑥4
2) state whether the function is a polynomial if so write it in standard form then state the degree,
type, leading coefficient and the constant term term a) 28)(f xx b) 386)(f 4 xxx
c) 6)(g 4 xx d) 1510)(h 23 xxx e) 103)(h 3
2
5 xxx f) xxxx 223 48)(g
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3) Use direct substitution to evaluate the polynomial function for the given value of x term
a) 1,151025)(f 23 xxxxx b) 2;358)(f 324 xxxxxx
c) 3;24)(g 53 xxxx d) 5;20256)(h 3 xxxx
e) 4;10)(h 3
4
34
2
1 xxxxx f) 2;51064)(g 235 xxxxxx
4) Use synthetic substitution to evaluate the polynomial function for the given value of x
a) 3;16825)(f 23 xxxxx b) 2;9563128)(f 2234 xxxxxxx
c) 6;3578)(g 23 xxxxx d) 4;35148)(h 3 xxxx
e) 2;23813)(h 43 xxxxx f) 3;27106)(g 35 xxxx
g) 3;7114)(h 32 xxxx h) 4;203)(h 4 xxxx
5) Describe the degree and the leading coefficient for the polynomial function graphed below
6) The weight of an ideal round-cut
diamond can be modelled by 𝑤 =0.0071𝑑3 − 0.090𝑑2 + 0.48𝑑where w
is the diamond’s weight (in carats) and d
is the diameter (in mm). according to the
model what is the weight of a diamond
with a diameter of 15mm
7) From 1992 to 2003, the number of people in the United States who participated in
skateboarding can be modelled by 5552.062.014.00076.0)(S 234 ttttt where S is
the number of participants (in millions) and t is the number of years since 1992. Graph the
model hence estimate the first year that the number of skateboarding participants was greater
than 8 million.
8) A cubic polynomial function f has leading coefficient 2 and constant term -5. If f(1)=0 and
f(2)=3, find f(-5)
9) From 1987 to 2003, the number of indoor movie screens M in the United States can be
modelled by 32 11267592600,21)(M tttt where t is the number of years since 1987.
a) State the degree and type of function
b) Make a table of values for the function hence graph it
10) From 1992 to 2003, the number of people in the United States who participated in
snowboarding can be modelled by 2.1037.084.0021.00013.0)(S 234 ttttt where S is
the number of participants (in millions) and t is the number of years since 1992. Graph the
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model hence estimate the first year that the number of snowboarding participants was greater
than 2 million.
11) From 1980 to 2002, the number of quarterly periodicals P published in the United States can
be modelled by 4502398.8624.0138.0)(P 234 ttttt where t is the number of years
since 1980.
a) Describe the end behavior of the graph of the model.
b) Graph the model on the domain 220 t
c) Use your graph to estimate the number of quarterly periodicals in the year 2010. Is it
appropriate to make these predictions?
12) The weights of Sarus crane chicks S and hooded crane chicks H (both in grams) during the
10 days for following hatching can be modelled by the functions
1366.1449.3122.0 23 tttS and 1246.20471.3115.0 23 tttH where t is the
number of days after hatching.
a) According to the models what is the difference in weight between 5-day old Sarus crane
chicks and hooded crane chicks?
b) Graph the two models
c) A biologist finds that the weight of crane chick 3 days after hatching is 130 games. What
species of crane is the chick more likely to be? Explain how you arrived at that answer.
13) The weight y (in pounds) of a rainbow trout can be modelled by 3000304.0 xy where x is
the length of the trout (in inches)
a) Write a function that relates the weight y and length x of a rainbow trout if y is measured
in kilograms and x is measured in centimeters. Use the fact that 1kg 20.2 pounds and
1cm 394.0 inches
b) Graph the original function and the function from part (a) on the same coordinate plane.
What type of transformation can you apply to the graph 3000304.0 xy to produce the
graph from part (a)?
Finding Rational Zeros of polynomials
The polynomial function 10534120.64)(f 23 xxxx has 8
7
4
5
2
3 and as its zeros.
Notice that the numerators of these zeros (-3, -5, and 7) are factors of the constant
term, -105. Also notice that the denominators (2, 4 and 8) are factors of the leading coefficient,
64. These observations are generalized by the rational zero theorem.
The Rational Zero Theorem
If 0
2
2
1
1 .....)(f axaxaxax n
n
n
n
has integer coefficients, then every rational zero of ƒ
has the following form:na
a
q
p
t coefficien leading offactor
ermconstant t offactor 0
Example 1 list all the rational zeros of;
a) 20118.)(f 23 xxxx b) 1093..4)(f 234 xxxxx
Solution
a) Factors of the constant term; 20,10,5,4,2,1 Factors of the leading coefficient; 1
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Possible rational zeros; 1
20,
1
10,
1
5,
1
4,
1
2,
1
1x
b) Factors of the constant term; 10,5,2,1 Factors of the leading coefficient; 4,2,1
Possible rational zeros; 4
5,
4
1,
2
5,
2
1,
1
10,
1
5,
1
2,
1
1x
Example 2 Find the rational zeros o 12112.)(f 23 xxxx
Solution
List the possible rational zeros. The leading coefficient is 1 and the constant term is º12. So, the
possible rational zeros are: 1
12
1
6
1
43
1
2
1
1x
Test x = 1: 20121121)1(f Test x =-1: 0121121)1(f
We can also Test these zeros using synthetic division as shown below
Since -1 is a zero of f, you can write )12)(1(121142.)(f 223 xxxxxxx
Factor the trinomial and use the factor theorem. )4)(3)(1.()12)(1.()(f 2 xxxxxxx
The zeros of f are -1, 3, and -4
In Example 1, the leading coefficient is 1. When the leading coefficient is not 1, the list of
possible rational zeros can increase dramatically. In such cases the search can be shortened by
sketching the function’s graph—either by hand or by using a graphing calculator
Example 3 Find all real zeros of 12529310)(f 234 xxxxx
Solution
The possible rational zeros of f are;
10
3
10
1
5
12
5
6
5
4
5
3
5
2
5
1
2
3
2
1
1
12
1
6
1
4
1
3
1
2
1
1x
Choose values to check. With so many
possibilities, it is worth your time to sketch
the graph of the function. From the graph, it
appears that some reasonable choices are
2
3 and
5
4
5
3
2
3 xxxx
Check these values using direct substitution
Test 2
3x : 012520310f2
32
2
33
2
34
2
3
2
3 Factor out a binomial 23x
)495)(32()821810)((12520310.)(f 2323
2
3234 xxxxxxxxxxxxx
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Repeat the steps above for 495)(g 23 xxxx any zero of g will also be a zero of ƒ. The
possible rational zeros of g are 5
4
5
2
5
1
1
4
1
2
1
1x .
The graph of f shows that 5
4 may be a zero. Now 0495g
5
42
5
43
5
4
5
4 Factor out a
binomial 5
4x So )1)(45)(32()555)()(32()(f 22
5
4 xxxxxxxxx
Find the remaining zeros of f by using the quadratic formula to solve 012 xx
The real zeros of f are; 2
51 and
2
51
5
4,
2
3
xxxx
Example 4 You are designing a candle-making kit. Each kit will contain 25 cubic inches of
candle wax and a mold for making a model of the pyramid-shaped building at the Louvre
Museum in Paris, France. You want the height of the candle to be 2 inches less than the length of
each side of the candle’s square base. What should the dimensions of your candle mold be?
Solution
The volume is BhV3
1 where B is the area of the base and h is the height. Let the side of
square base be x inches then height is x-2 and since V=25 then
0752275)2(25 23232
3
1 xxxxxx
The possible rational solutions are 752515531 x . Use the possible solutions.
Note that in this case, it makes sense to test only positive x-values
So x = 5 is a solution. The other two solutions, which satisfies 01532 xx are
2
513
x and can be discarded because they are imaginary.
The base of the candle mold should be 5 inches by 5 inches. The height of the mold should be
5 − 2 = 3 inches.
Example 5 Some ice sculptures are
modelled by filling a mold with water and
then freezing it. You are making such an ice
sculpture for a school dance. It is toi be
shaped like a pyramid with height that is 1
foot greater than the length of each side of
the square base. The volume of the ice
sculpture is 4 cubic feet. What are the
dimensions of the mold?v
Solution
Write an equation for the volume; )1(4 height area) (base= volume 2
3
1
3
1 xx
Thus 012- 12 2323 xxxx . List the possible rational zeros
1
12
1
6
1
4
1
3
1
2
1
1x Test possible solutions only positive values make sense
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2 x is a root of the function. Thus 0)63x)(2(12- 223 xxxx . The other 2 solutions
that satisfies 063x2 x are imaginary therefore they can be discarded. The only reasonable solution
is 2x the base of the mold is 2 feet by 2 feet and the height is 3 feet.
Modeling with Polynomial Functions
You know that two points determine a line and that three points determine a parabola. In
Example 1 you will see that four points determine the graph of a cubic function.
Example 1 Write the cubic function whose graph is shown at the right.
Solution
Use the three given x-intercepts to write the
following: ƒ(𝑥) = 𝑎(𝑥 + 3)(𝑥 − 2)(𝑥 − 5)
To find a, substitute the coordinates of the
fourth point. −15 = 𝑎(0 + 3)(0 − 2)(0 −
5), 𝑠𝑜 𝑎 = −0.5
Check the graph’s end behavior. The degree
of ƒ is odd and 0na , so )(f x as
x and )(f x as x .
To decide whether y-values for equally-
spaced x-values can be modeled by a
polynomial function, you can use finite
differences.
Example 2 The first three triangular numbers are shown at the right.
A formula for the nth triangular number is
ƒ(𝑛) = 1 2 (𝑛2 + 𝑛). Show that this
function has constant 2nd order differences. Solution
Write the first several triangular numbers. Find the first-order differences by subtracting
consecutive triangular numbers. Then find the second-order differences by subtracting
consecutive first-order differences.
Each 2nd order difference is 1, so the 2nd order difference is a constant.
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Example 3 The profit P (in $”000,000) for a shoe manufacturer can be modeled by
xxP 4621 3 where x is the number of shoe produced (in millions). The company now
produces 1 million shoes and makes a profit if $25,000,000, but would like to cut back
production. What lesser number of shoe could the company produce and still make the same
profit?
Solution
Substitute 25 for P t0 get 0254621462125 33 xxxx . But 1x is one solution of
the equation 1 x is a factor of 0254621 3 xx . Use synthetic division to get the other
factors. So )252121)(1(254621 23 xxxxx .
Use quadratic formula to find that 7.0x is
the other positive solution.
The company could still make the same
profit producing about 700,000 shoe
Exercise
1) List the possible rational zeros of ƒ using the rational zero theorem.
a) f(𝑥) = 𝑥3 + 14𝑥2 + 41𝑥 − 56
b) f(𝑥) = 𝑥3 − 17𝑥2 + 54𝑥 + 72
c) f(𝑥) = 2𝑥3 + 7𝑥2 − 𝑥 + 30
d) f(𝑥) = 5𝑥4 + 12𝑥3 − 16𝑥2 + 10
e) f(𝑥) = 𝑥4 + 2𝑥2 − 24
f) f(𝑥) = 2𝑥3 + 5𝑥2 − 6𝑥 − 1
g) f(𝑥) = 2𝑥5 + 𝑥2 + 16
h) f(𝑥) = 2𝑥3 + 9𝑥2 − 53𝑥 − 60
i) f(𝑥) = 6𝑥4 − 3𝑥3 + 𝑥 + 10
j) f(𝑥) = 4𝑥3 + 5𝑥2 − 3
k) f(𝑥) = 8𝑥2 − 12𝑥 − 3
l) f(𝑥) = 3𝑥4 + 2𝑥3 − 𝑥 + 15
2) For each polynomial function, decide whether you can use the rational zero theorem to find
its zeros. Explain why or why not. (a) f(𝑥) = 6𝑥2 − 8𝑥 + 4 b) f(𝑥) = 0.3𝑥2 + 2𝑥 + 4.5
c) f(𝑥) =1
4𝑥2 − 𝑥 +
7
8
3) Find all the real zeros of the function
a. f(𝑥) = 𝑥3 − 3𝑥2 − 6𝑥 + 8
b. f(𝑥) = 𝑥3 + 4𝑥2 − 𝑥 − 4
c. f(𝑥) = 2𝑥3 − 5𝑥2 − 2𝑥 + 5
d. f(𝑥) = 2𝑥3 − 𝑥2 − 15𝑥 + 18
e. f(𝑥) = 𝑥3 + 4𝑥2 + 𝑥 − 6
f. f(𝑥) = 𝑥3 + 5𝑥2 − 𝑥 − 5
4) Suppose you have 18 cubic inches of wax and you want to make a candle in the shape of a
pyramid with a square base. If you want the height of the candle to be 3 inches greater than
the length of each side of the base, what should the dimensions of the candle be?
5) Use synthetic division to decide which of the values 1, -1, 2, and -2 are zeros of the function
a) f(𝑥) = 𝑥3 + 7𝑥2 − 4𝑥 − 28
b) f(𝑥) = 𝑥3 + 5𝑥2 + 2𝑥 − 8
c) f(𝑥) = 𝑥4 + 3𝑥3 − 7𝑥2 − 27𝑥 − 18
d) f(𝑥) = 2𝑥4 − 9𝑥3 + 8𝑥2 + 9𝑥 − 10
e) f(𝑥) = 𝑥4 + 𝑥3 + 3𝑥2 − 3𝑥 − 4
f) f(𝑥) = 3𝑥4 + 3𝑥2 + 2𝑥2 + 5𝑥 − 10
g) f(𝑥) = 𝑥3 − 3𝑥2 + 4𝑥 − 12
h) f(𝑥) = 𝑥3 + 𝑥2 − 11𝑥 + 10
i) f(𝑥) = 𝑥6 − 2𝑥4 − 11𝑥 + 12
j) f(𝑥) = 𝑥5 − 𝑥4 − 2𝑥3 − 𝑥2 + 𝑥 + 2
6) Find all the real zeros of the function.
a) f(𝑥) = 𝑥3 − 8𝑥2 − 23𝑥 + 30
b) f(𝑥) = 𝑥3 + 2𝑥2 − 11𝑥 − 12
c) f(𝑥) = 𝑥3 − 7𝑥2 + 2𝑥 + 40
d) f(𝑥) = 𝑥3 + 𝑥2 − 𝑥 − 2
e) f(𝑥) = 𝑥3 + 72 − 5𝑥2 − 18𝑥
f) f(𝑥) = 𝑥3 + 9𝑥2 − 4𝑥 − 36
g) f(𝑥) = 𝑥4 − 5𝑥3 + 7𝑥2 + 3𝑥 − 10
h) f(𝑥) = 𝑥4 + 𝑥3 + 𝑥2 − 9𝑥 − 10
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Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 100
i) f(𝑥) = 𝑥4 + 𝑥3 − 11𝑥2 − 9𝑥 + 18
j) f(𝑥) = 𝑥4 − 3𝑥3 + 6𝑥2 − 2𝑥 − 12
k) f(𝑥) = 𝑥5 + 𝑥4 − 9𝑥3 − 5𝑥2 − 36
l) f(𝑥) = 𝑥5 − 𝑥4 − 7𝑥3 + 11𝑥2 −
8𝑥 + 12
7) Find the real zeros of the function. Then match each function with its graph.
(a) f(𝑥) = 𝑥3 + 2𝑥2 − 𝑥 − 2 (b) f(𝑥) = 𝑥3 − 3𝑥 − 2 (c) f(𝑥) = 𝑥3 − 𝑥2 + 2
8) Use the graph to shorten the list of possible rational zeros. Then find all the real zeros of the
function.
a) f(𝑥) = 4𝑥3 − 12𝑥2 − 𝑥 + 15
b) f(𝑥) = −3𝑥3 + 20𝑥2 − 36𝑥 + 16
9) Find all the real zeros of the function.
a) f(𝑥) = 2𝑥3 + 4𝑥2 − 2𝑥 − 4
b) f(𝑥) = 2𝑥3 − 5𝑥2 − 14𝑥 + 8
c) f(𝑥) = 2𝑥3 − 5𝑥2 − 𝑥 + 6
d) f(𝑥) = 2𝑥3 + 𝑥2 − 50𝑥 − 25
e) f(𝑥) = 2𝑥3 − 𝑥2 − 32𝑥 + 16
f) f(𝑥) = 3𝑥3 + 12𝑥2 + 3𝑥 − 18
g) f(𝑥) = 2𝑥4 + 3𝑥3 − 3𝑥2 + 3𝑥 − 5
h) f(𝑥) = 3𝑥4 − 8𝑥3 − 5𝑥2 + 16𝑥 − 5
i) f(𝑥) = 2𝑥4 + 𝑥3 − 𝑥2 − 𝑥 − 1
j) f(𝑥) = 3𝑥4 + 11𝑥3 + 11𝑥2 + 𝑥 − 2
k) f(𝑥) = 2𝑥5 + 𝑥4 − 32𝑥 − 16
l) f(𝑥) = 3𝑥5 + 𝑥4 − 243𝑥 − 18
10) From 1990 to 1994, the mail order sales of health products in the United States can be
modeled by 𝑆 = 10𝑡3 + 115𝑡2 + 25𝑡 + 2505 where S is the sales (in millions of dollars)
and t is the number of years since 1990. In what year were about $3885 million of health
products sold? (Hint: First substitute 3885 for S, then divide both sides by 5.)
11) The profit P (in millions of dollars) for a manufacturer of MP3 players can be modeled by 32 41216 xxxP where x is the number of MP3 players produced (in millions). Currently
the company produces 3 million MP3 players and makes a profit if $48,000,000. What lesser
number of MP3 players could the company produce and still make the same profit?
12) You have a picture that you want to frame, but first you have to put a mat around it. The
picture is 12 inches by 16 inches. The area of the mat is 204 square inches. If the mat extends
beyond the picture the same amount in each direction, what will the final dimensions of the
picture and mat be?
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Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 101
13) You are designing a monument and a
base as shown at the right. You will use
90 cubic feet of concrete for both pieces.
Find the value of x.
14) At a factory, molten glass is poured into
molds to make paperweights. Each mold
is a rectangular prism whose height is 3
inches greater than the length of each
side of the square base. A machine pours
20 cubic inches of liquid glass into each
mold. Find the dimensions of the mold.
15) You are building a solid concrete wheelchair ramp. The width of the ramp is three times the
height, and the length is 5 feet more than 10 times the height. If 150 cubic feet of concrete is
used, what are the dimensions of the ramp?
16) You are designing a kit for making sand
castles. You want one of the molds to be
a cone that will hold 48π cubic inches of
sand. What should the dimensions of the
cone be if you want the height to be 5
inches more than the radius of the base?
17) You are designing an in-ground lap
swimming pool with a volume of 2000
cubic feet. The width of the pool should
be 5 feet more than the depth, and the
length should be 35 feet more than the
depth. What should the dimensions of
the pool be?
18) You are a landscape artist designing a patio. The square patio floor is to be made from 128
cubic feet of concrete. The thickness of the floor is 15.5 feet less than each side length of the
patio. What are the dimensions of the patio floor?
19) At a factory molten glass is poured into a mold to make paperweights. Each model is a
rectangular prism with a height 4 inches greater than the length of each side of the square
base. Each mold holds 63 cubic inches of molten glass. Find the dimensions of the mold.
20) Write a polynomial equation to model the situation then list the possible rational zeros of the
fequation
a) A rectangular prism has edges of length x, x-1 and x-2 and a volume of 24.
b) A pyramid has a square base with sides of length x, a height of 2x-5 and a volume of 3.
21) From 1994 t0 2003, the amount of athletic equipment E (in millions of dollars) sold
domestically can be modeled by 32 1014020150,18)(E tttt where t is the number of
years since 1994. Use the following steps to find the year when about $20,300 million of
athletic equipmen was sold.
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a) Write a polynomial equation that can be used to find the answer.
b) List the possible whole number solutions of the equation in part a) that are <10
c) Use synthetic division to determine which of the possible solutions in part b) is an actual
solution, then calculate the year which corresponds to the solution.
22) From 1990 t0 2000, the amount of U.S travelers to foreigh countries F (in 1000’s) can be
modeled by 916,433924202826412)(F 234 ttttt where t is the number of years
since 1990. Use the following steps to find the year when there were about 56,300,000
travelers.
a) Write a polynomial equation that can be used to find the answer.
b) List the possible whole number solutions of the equation in part a) that are 10
c) Use synthetic division to determine which of the possible solutions in part b) is an actual
solution.
d) Graph the function F(t) and explain why there are no other reasonable solutions then
calculate the year which corresponds to the solution.
23) The profit P (in millions of dollars) for a T-shirt manufacturer can be modeled by 324 xxxP where x is the number of shoe produced (in millions). Currently the
company produces 4 million T-shirts and makes a profit if $4,000,000. What lesser number
of T-shirt could the company produce and still make the same profit?
24) From 1985 to 2003, the total attendace A (in thausands) at NCAA women’s basketball games and the
number Tof NCAA women’s basketball teams can be modeled by the function 32 95.11.701882150 xxxA and
32 95.11.708.14725 xxxA where x is the
number of years since 1985. Write a function for the average attendance per team from 1985
to 2003.
25) The price P (in dollars) that a radio manufacturer is able to charge for a radio is given by 2440 xP where x is the number (in millions) of radios produced.it costs the company
$15 to make a radio.
a) Write an expression for the company’s total revenue in terms of x
b) Write a function for the company’s profit P by subtracting the total cost of making x
radios from the expression in part a)
c) Currently, the company produces 1.5 million radios and makes a profit if $24,000,000.
Write and solve an equation to find a lesser number of radios that he company could
tproduce and still make the same profit.
d) Do all the solutions in part a) make sense in this situation? explain
26) From 1990 to 2000, over night stays S and the total visits V (both in millions) to national
parks can be modeled by 432 0072.0176.040.139.36.17 xxxxS and
25610.3 xV where x is the number of years since 1990. Write a function for percent of
visits to national parks that are over night stays. Explain how yiou constracted your function.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
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Conic Sections Parabolas, circles, ellipses, and hyperbolas are all curves that are formed by the intersection of a
plane and a double-napped cone. Therefore, these shapes are called conic sections or simply
conic. The equation of any conic can be written in the form 𝐴𝑥2 + 𝐵𝑥𝑦 + 𝐶𝑦2 + 𝐷𝑥 + 𝐸𝑦 +
𝐹 = 0 which is called a general second-degree equation in x and y. The expression 𝐵2 − 4𝐴𝐶
is called the discriminant of the equation and can be used to determine which type of conic the
equation represents.
History of Conics
In 200 B.C conic sections were studied thoroughly for the first time by a Greek mathematician
named Apollonius. Six hundred years later, the Egyptian mathematician Hypatia simplified the
works of Apollonius, making it accessible to many more people. For centuries, conics were
studied and appreciated only for their mathematical beauty rather than for their occurrence in
nature or practical use. TODAY astronomers know that the paths of celestial objects, such as
planets and comets, are conic sections. For example, a comet’s path can be parabolic, hyperbolic,
or elliptical.
Circle
A Circle is the set of all points (x, y) that are equidistant from a fixed point, called the center of
the circle. The distance r between the center and any point (x, y) on the circle is the radius. The
distance formula can be used to obtain an equation of the circle whose center is the origin and
whose radius is r. Because the distance
between any point (x, y) on the circle and
the center (0, 0) is r, you can write the
following.
22222 )0()0( ryxyxr
The standard form of the equation of a circle
with center at (0, 0) and radius r is as 222 ryx Eg A circle with center at (0, 0)
and radius 3 has equation 922 yx
Example 1 Draw the circle given by 22 25 xy .
Solution
Write the equation in standard form.
2525 2222 yxxy
In this form you can see that the graph is a
circle whose center is the origin and whose
radius is 5
Plot several points that are 5 units from the
origin. The points (0, 5), (5, 0), (0, º5), and
(º5, 0) are most convenient. Draw a circle
that passes through the four points.
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Example 2 The point (1, 4) is on a circle whose center is the origin. Write the standard form of
the equation of the circle.
Solution
Because the point (1, 4) is on the circle, the radius of the circle must be the distance between the
center and the point (1, 4) 17)04()01( 222r the equation of the circle is 1722 yx .
A theorem in geometry states that a line
tangent to a circle is perpendicular to the
circle’s radius at the point of tangency. In
the diagram, AB is tangent to the circle
with center C at the point of tangency B, so
AB is perpendicular to BC .
Example 3 Write an equation of the line that is tangent to the circle 1322 yx at (2, 3).
Solution
The slope of the radius through the point (2,
3) is 2
3
02
03
m
Because the tangent line at (2, 3) is
perpendicular to this radius, its slope must
be the negative reciprocal of 2
3 , or
3
2 .
So, an equation of the tangent line
1332)2(3
23 yxxy .
Translated Circle
The standard equation of a translated circle center (h , k) is of the form (𝑥– ℎ)2 + (𝑦– 𝑘)2 = 𝑟2
where r is the radius of a circle. To write the standard equation of a translated circle
i) Group x terms together, y-terms together, and move constants to the other side
ii) Complete the square for the x-terms and for the y-terms
Remember that whatever you do to one side, you must also do to the other
Example 1 Find the equation of a circle of radius10 centered at (16, 10)
Solution The equation is(𝑥– 16)2 + (𝑦– 10)2 = 100
Example 2 Obtain the standard equation of the circle 𝑥2 + 𝑦2 − 10𝑥 + 8𝑦 − 8 = 0 hence find
the center and radius
Solution
49))4()5(16258)168()2510(8)8()10( 222222 yxyyxxyyxx
Therefore the center is (5 , -4) and the radius 𝑟 = 7
Question Obtain the standard equation of the circle 𝑥2 + 𝑦2 − 10𝑥 + 8𝑦 − 8 = 0 hence find
the center and radius 𝑥2 + 𝑦2 + 6𝑥 − 12𝑦 + 20 = 0
Example 3 Find the center, the radius, and write the standard equation of the circle if the
endpoints of a diameter are (-8 , 2) and (2 , 0).
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Solution
Center: (−8+2
2 ,
2+0
2) = (−3 , 1) Using midpoint formular
Radius: using distance formula with radius and an endpoint √((2 − (−3))2 + (0 − 1))2 = √26
Thus the equation is (𝑥 + 3)2 + (𝑦 − 1)2 = 26
Question Obtain the standard equation of the circle 4𝑥2 + 4𝑦2 − 16𝑦 + 8𝑥 − 50 = 0 hence
find the center and radius
Example 4 A cellular phone transmission tower located 10 miles west and 5 miles north of
your house has a range of 20 miles. A second tower, 5 miles east and 10 miles south of your
house, has a range of 15 miles.
a) Write an inequality that describes each tower’s range.
b) Do the two regions covered by the towers overlap?
Solution
a) Let the origin represent your house. The first tower is at (-10, 5) and the boundary of its
range is a circle with radius 20. Substitute -10 for h, 5 for k, and 20 for r into the standard
form of the equation of a circle. (𝑥 + 10)2 + (𝑦 − 5)2 = 400 . The second tower is at (5, -
10). The boundary of its range is a circle with radius 15. (𝑥 − 5)2 + (𝑦 − 10)2 = 225
b) One way to tell if the regions overlap is
to graph the inequalities. You can see
that the regions do overlap. You can also
check whether the distance between the
two towers is less than the sum of the
ranges.
35215
1520))10(5()310( 22
Parabolas
A parabola is the set of all points in the
plane equidistant from a fixed point P
(called the focus) and a fixed line l (called
the directrix). The standard equation of a
Parabola with a vertical axis and a vertex at
the origin is given by 𝑥2 = 4𝑝𝑦. The
standard equation of a Parabola with a
horizontal axis and a vertex at the origin is
given by 𝑦 2 = 4𝑝𝑥
A parabolas can have a vertical axis of symmetry implying it can either open up or down or a
horizontal axis of symmetry implying it can open left or right. See the four cases shown below,
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Example 1 Identify the focus and directrix of the parabola given by 𝑦2 = −6𝑥 . Draw the
parabola.
Solution
Because the variable y is squared, the axis of symmetry is horizontal.
To find the focus and directrix, rewrite the
equation as 𝑦2 = 4(−1.5)𝑥. The focus is
(-1.5, 0) and the directrix is 𝑥 = 1.5 To
draw the parabola, make a table of values
and plot points. Because 𝑝 < 0, the parabola
opens to the left. Therefore, only negative x-
values should be chosen.
X -1 -2 -3 -4 -5
Y
±2.4
5
±3.4
6
±4.2
4
±4.9
0
±5.4
8
Example 2 Write an equation of the parabola shown at the right.
Solution
The graph shows that the vertex is (0, 0) and
the directrix is 𝑦 = −𝑝 = −2. Substitute 2
for p in the standard equation for a parabola
with a vertical axis of symmetry.
𝑥2 = 4𝑝𝑦 Standard form, Substitute 𝑝 = 2
and Simplify ie 𝑥2 = 4(2)𝑦 = 8𝑦
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Example 3 Sunfire is a glass parabola used to collect solar energy. The sun’s rays are reflected
from the mirrors toward two boilers located at the focus of the parabola. When heated, the
boilers produce steam that powers an alternator to produce electricity.
a) Write an equation for Sunfire’s cross section.
b) How deep is the dish?
Solution
a) The boilers are 10 feet above the vertex of the dish. Because the boilers are at the focus and
the focus is p units from the vertex, you can conclude that p = 10. Assuming the vertex is at
the origin, an equation for the parabolic cross section is 𝑥2 = 4𝑝𝑦 substituting p=10 we get
is 𝑥2 = 40𝑦
b) The dish extends is 37
2= 18.5 feet on either side of the origin. To find the depth of the dish,
substitute 18.5 for x in the equation from part (a). (18.5)2 = 40𝑦this gives 𝑦 = 8.6 The dish
is about 8.6 feet deep.
Translated Parabola
The general equation of a translated Parabola with a vertical axis is given by
(𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘)
The vertex is at (h, k), and the focus at (h, k + p). The graph is a parabola which opens up or
down, depending on the sign of p.
The general equation of a translated Parabola with a horizontal axis is given by
(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)
The vertex is at (h, k), and the focus at (h + p, k). The graph is a parabola which opens right or
left, depending on the sign of p
Example 1 Write an equation of the parabola whose vertex is (-2, 1) and whose focus is (-3, 1).
Solution
Choose form: Begin by sketching the
parabola, as shown. Because the parabola
opens to the left, it has the form
(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ) where p < 0. Find
h and k: The vertex is at (-2, 1), so
ℎ = −2 and 𝑘 = 1. Find p: The distance
between the vertex (-2, 1) and the focus
(-3, 1) is 𝑝 = −1. The standard form of the
equation is (𝑦 − 1)2 = 4𝑝(𝑥 + 2)
Example 2 find the equation of the parabola whose focus is (3 , 4) and directrix is 𝑥 = 1
Solution
The vertex is halfway between the focus and directrix, at (2 , 4)
The parabola opens to the right, so the general form is (𝑦 − 4)2 = 4(𝑥 − 2)
In the special case where the focus is on the directrix, then the resulting parabola i
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Example 3 Write the equation of a parabola with a vertex of ( –2, 4) and focus point (0,
4). Also sketch the parabola.
Solution
Example 4 Given the equation of the parabola 4𝑥2 − 24𝑥 − 40𝑦 − 4 = 0 From the Standard
Form, determine the co-ordinates of the vertex and the equation of the directrix.
Solution
The equation can be written as 440)6(40440244 22 yxxyxx
)1(4)1(10)3(
4040)3(4
2
52
2
yyx
yx
Vertex: (ℎ, 𝑘) = (3, −1) , Since ‘x’ is
squared, this parabola opens vertically.
Since ‘p’ is positive, it opens upward. 𝑝 =
2.5 → Focus: (3, 1.5) The Directrix is a
horizontal line below the parabola, ‘p’ units
from the vertex. Directrix: 𝑦 = −3.5 The
Parabola opens upward from the Vertex,
away from the Directrix, around the Focus.
Example 3 Determine the co-ordinates of the vertex and the equation of the directrix for the
parabola 𝑦2 + 2𝑥 = 0
Solution
𝑦2 = 4(−0.5)𝑥 Vertex: (h, k) = (0, 0) Since
‘y’ is squared, this parabola opens
horizontally. Since ‘p’ is negative, it opens
to the left. 𝑝 − 0.5 Focus: (−0.5, 0) The
Directrix is a vertical line to the right of the
parabola, ‘p’ units from the vertex.
Directrix: 𝑥 = 0.5 The Parabola opens to the
left from the Vertex, away from the
Directrix, around the Focus.
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Exercise
1) Graph the equation. Identify the focus and directrix of the parabola a) yx 42 b) 26xy
c) 0122 xy d) yx 43 2 e) 0282 yx f) 049 2 xy g) 02
8
1 yx h)
02
20
1 yx i) 160208 2 xy
2) Graph the parabola 4(𝑦 + 2) − (𝑥 − 1)2 = −4
3) Tell whether the parabola opens up, down, left, or right. (a) 23xy (b) yx 29 2 (c)
xy 62 2 (d) 27yx (e) yx 162 (f) xy 83 2 (g)
25 yx (h) yx3
42
4) Write the standard form of the equation of the parabola with the given focus or directrix and
vertex at (0, 0).
Focus; i) (-2 , 0) ii)(0 , -0.5) iii) (0 , 8
3) b) Directrix; i) 4x ii) 2
5y iii) 02
1 x
5) Match the equation with its graph;
a) xy 42 b) yx 42 c) yx 42 xy 42 d) e) xy4
12 f) yx4
12
6) Find the equation of the parabola whose vertex is at (−7, 2) and whose focus is at (−4, 2)
7) Graph the parabola and identify the focus, axis of symmetry, the directrix, and the focal
diameter: a) 8𝑥2 − 24𝑦 = 0 b) 6𝑦 2 − 24𝑥 = 0 c) 𝑦 = −2𝑥2 − 12𝑥– 19
8) The cross section of a television antenna
dish is a parabola. For the dish at the
right, the receiver is located at the focus,
4 feet above the vertex. Find an equation
for the cross section of the dish.
(Assume the vertex is at the origin.) If
the dish is 8 feet wide, how deep is it?
9) A searchlight has a parabolic reflector (has a cross section that forms a “bowl”). The
parabolic “bowl” is 16 inches wide from rim to rim and 12 inches deep. The filament of the
light bulb is located at the focus.
a) What is the equation of the parabola used for the reflector?
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b) How far from the vertex is the filament of the light bulb?
10) The filament of a lightbulb is a thin wire that glows when electricity passes through it.
11) The filament of a car headlight is at the
focus of a parabolic reflector, which
sends light out in a straight beam. Given
that the filament is 1.5 inches from the
vertex, find an equation for the cross
section of the reflector. If the reflector is
7 inches wide, how deep is it?
12) In the drawing shown at the left, the rays
of the sun are lighting a candle. If the
candle flame is 12 inches from the back
of the parabolic reflector and the
reflector is 6 inches deep, then what is
the diameter of the reflector?
13) The cables of the middle part of a
suspension bridge are in the form of a
parabola, and the towers supporting the
cable are 600 feet apart and 100 feet
high. What is the height of the cable at a
point 150 feet from the center of the
bridge?
14) For an equation of the form2axy , discuss what effect increasing |a| has on the focus and
directrix.
15) You can make a solar hot dog cooker
using foil-lined cardboard shaped as a
parabolic trough. The drawing at the
right shows how to suspend a hot dog
with a wire through the focus of each
end piece. If the trough is 12 inches wide
and 4 inches deep, how far from the
bottom should the wire be placed?
16) A flashlight has a parabolic reflector. An equation for the cross section of the
reflector is xy7
322 . The depth of the reflector is 2
3 inches.
Explain why the value of p must be less than the depth of the reflector of a flashlight.
How wide is the beam of light
projected by the flashlight?
Write an equation for the cross
section of a reflector having the same
depth but a wider beam than the
flashlight shown. How wide is the
beam of the new reflector?
Write an equation for the cross section of a reflector having the same depth but a
narrower beam than the flashlight shown. How wide is the beam of the new reflector?
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17) The latus rectum of a parabola is the line
segment that is parallel to the directrix,
passes through the focus, and has
endpoints that lie on the parabola. Find
the length of the latus rectum of a
parabola given by pyx 42 .
Ellipses An ellipse is the locus of all points pthe sum of whose distances from 2 ixed points called foci is
constant. 𝑑1 + 𝑑2 = 2𝑎 The line through the foci intersects the ellipse at two points, the vertices.
The line segment joining the vertices is the
major axis, and its midpoint is the center of
the ellipse. The line perpendicular to the
major axis at the center (called minor axis
which has length 2b) intersects the ellipse at
two points called the co-vertices..
The two types of ellipses we will discuss are those with a horizontal major axis and those with a
vertical major axis
The standard form of the equation of an ellipse with center at (0, 0) and major and minor axes of
lengths 2a and 2b, where 𝑎 > 𝑏 > 0, is as follows.
Equation Major axis Vertex Co-vertex
12
2
2
2
b
y
a
x
Horizontal (±𝑎 , 0) (0 , ±𝑏 )
12
2
2
2
a
y
b
x
Vertical (0 , ±𝑎 ) (±𝑏 , 0)
The foci of the ellipse lie on the major axis, c units from the center where 𝑐2 = 𝑎2 − 𝑏2
Example 1 Draw the ellipse given by 9𝑥2 + 16𝑦2 = 144. Identify the foci.
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Solution
First rewrite the equation in standard form 9𝑥2 + 16𝑦2 = 144 1916
22
yx
Because the denominator of the 𝑥2 -term is
greater than that of the 𝑦2 -term, the major
axis is horizontal. So, 𝑎 = 4 and 𝑏 = 3. Plot the vertices and co-vertices. Then draw
the ellipse that passes through these four
points. The foci are at (c, 0) and (-c, 0). To
find the value of c, use the equation
𝑐2 = 𝑎2 − 𝑏2 = 16 − 9 = 7 .
The foci are at (±√(7 ,0)
Example 2 Write an equation of the ellipse with the given characteristics and center at (0, 0).
a)Vertex: (0, 7) Co-vertex: (-6, 0) b) Vertex: (-4, 0) Focus: (2, 0)
Solution
In each case, you may wish to draw the
ellipse so that you have something to check
your final equation against. a. Because the
vertex is on the y-axis and the co-vertex is
on the x-axis, the major axis is vertical with
a = 7 and b = 6.
Thus the equation is 14936
22
yx
Because the vertex and focus are points on a
horizontal line, the major axis is horizontal
with 𝑎 = 4 and 𝑐 = 2. To find b, use the
equation 𝑐2 = 𝑎2 − 𝑏2
1241162 b thus the equation is
11216
22
yx
Example 3 A portion of the White House lawn is called The Ellipse. It is 1060 feet long and
890 feet wide. a) Write an equation of The Ellipse. b) The area of an ellipse is 𝐴 = 𝜋𝑎𝑏. What
is the area of The Ellipse at the White House?
Solution
a) The major axis is horizontal with 𝑎 = 10 60 ÷ 2 = 530 and 𝑏 = 890 ÷ 2 = 445. Thus the
equation is 1445530 2
2
2
2
yx
. b) The area is A = π (530) (445) ≈ 741,000 square feet. Modeling
Remark: If 𝑎 = 𝑏 we get the equation of a circle
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Example 4 In its elliptical orbit, Mercury ranges from 46.04 million kilometers to 69.86
million kilometers from the center of the sun. The center of the sun is a focus of the orbit. Write
an equation of the orbit.
Solution
Using the diagram shown, you can write a
system of linear equations involving a and c.
𝑎 − 𝑐 = 46.04 𝑎 + 𝑐 = 69.86 Adding
the two equations gives 2𝑎 = 115.9, so
𝑎 = 57.95. Substituting this a-value into
the second equation gives 57.95 + c = 69.86,
so c = 11.91. From the relationship 𝑐2 = 𝑎2 − 𝑏2 , you can conclude the following:
An equation of the elliptical orbit is (57 x .9
2 5) 2 + (56 y .7 2 1) 2 = 1 where x and y are
in millions of kilometers.
Translated Ellipse
The standard equation of a translated ellipse centered at point (h , k) and major and minor axes of
lengths 2a and 2b, where 𝑎 > 𝑏 > 0 is as follows
Equation Major axis Vertex Co-vertex
1)()(
2
2
2
2
b
ky
a
hx
Horizontal
(ℎ± 𝑎 , 𝑘) (ℎ , 𝑘 ± 𝑏 )
1)()(
2
2
2
2
a
ky
b
hx
Vertical
(ℎ , 𝑘± 𝑎 ) (ℎ ± 𝑏 , 𝑘)
Example 1
Write an equation of the ellipse with foci at (3, 5) and (3, -1) and vertices at (3, 6) and (3, -2).
Solution
Plot the given points and make a rough sketch. The ellipse has a vertical major axis, so its
equation is of this form:
Find the center: The center is halfway between the vertices. (ℎ, 𝑘) = (3+3
2,
6−2
2) = (3, 2)
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Find a: The value of a is the distance
between the vertex and the center. 𝑎 =
√(3 − 3)2 + (6 − 2)2 = √02 + 42 = 4.
Find c: The value of c is the distance
between the focus and the center. 𝑐 =
√(3 − 3)2 + (5 − 2)2 = √02 + 32 = 3
Find b: Substitute the values of a and c into
the equation 𝑏2 = 𝑎2 − 𝑐2 = 16 − 9 = 7
The standard form of the equation is
Example 2 Identify the vertices, co-vertices, foci, and sketch the ellipses;
136
)2(
4
)3( 22
yx
Solution
Clearly 6a and 2b
32436 c . The center is (-3 ,
2)and the major axis is vertical the
major axis is 3x . The foci are at
(−3 , ±√32 ), the vertices are at (−3 , 8 )
and (−3 , −4 ) and the co-vertices are at
(−5 , 2) and (−1 , 2 )
The graph of the ellipse look like this
Exercise
1) Write an equation of an ellipse with the given characteristics and center (0 , 0)
a) a)Vertices:: (0, 5) co-vertices: (-4 , 0) d) Foci: )0,102( Vertices: (-7, 0)
b) Vertices:: (9, 0) co-vertices: (0 , 2) e) Co-vertices: )0,91( , Foci: (0 , 3)
c) Foci: (0 , -5) Vertices: (0, 13) f) Co-vertices: )33,0( , Foci: (4 , 0)
2) Graph the equation. Identify the vertices, co-vertices and the foci of the ellipse
(a) 13625
22
yx
b) 1494
22
yx
c) 1925
4 22
xx
d) 14
9
9
4 22
yx
e) 164
22
xy
f) 494
22
yx
g) 14
9
64
22
yx
h) 4595 22 yx i) 259 22 yx
3) Write the equation in standard form. Then identify the vertices, co-vertices, and foci of the
ellipse. a) 6416 22 yx b) 36412 22 yx c) 2502510 22 yx d) e) f)
4) Suppose a satellite’s orbit is an ellipse with Earth’s center at one focus. If the satellite’s least
distance from Earth’s surface is 150 miles and its greatest distance from Earth’s surface is
600 miles, write an equation for the ellipse. (Use 4000 miles as Earth’s radius.)
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5) An elliptical garden is 10 feet long and 6 feet wide. Write an equation for the garden. Then
graph the equation. Label the vertices, co-vertices, and foci. Assume that the major axis of
the garden is horizontal
6) Statuary Hall is an elliptical room in the United States Capitol in Washington, D.C.
The room is also called the Whispering
Gallery because a person standing at one
focus of the room can hear even a
whisper spoken by a person standing at
the other focus. This occurs because any
sound that is emitted from one focus of
an ellipse will reflect off the side of the
ellipse to the other focus. Statuary Hall
is 46 feet wide and 97 feet long.
a) Find an equation that models the
shape of the room.
b) How far apart are the two foci?
c) What is the area of the room’s floor?
7) The first artificial satellite to orbit Earth was Sputnik I, launched by the Soviet Union in
1957. The orbit was an ellipse with Earth’s center as one focus. The orbit’s highest point
above Earth’s surface was 583 miles, and its lowest point was 132 miles. Find an equation of
the orbit. (Use 4000 miles as the radius of Earth.) Graph your equation.
8) Australian football is played on an elliptical field. The official rules state that the field must
be between 135 and 185 meters long and between 110 and 155 meters wide. _Source: The
Australian News Network
a) Write an equation for the largest and for the smallest allowable playing field.
b) Write an inequality that describes the possible areas of an Australian football field.
9) A tour boat trvaels between two islands that are 12 miles apart. For a trip between the
islands, there is enough fuel for a 20-mile tour.
a) The region in which the boat can travel is bounded by an ellipse. Explain why this is so
b) Let (0, 0) be the center of the ellipse.
Find the coordinates of each island.
c) Suppose the boat travels from one
island, straight past the other island
to the vertex of the ellipse, and back
to the second island. How many
miles does the boat travel? What are
the coordinates of the vertex?
d) Use your answers to parts (b) and (c) to write an equation for the ellipse that bounds the
region the boat can travel in.
10) Show that 222 bac for any ellipse given by the equation 12
2
2
2
b
y
a
xwith foci at (c,0) and (-c, 0).
Hyperbolas
A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed
points is a constant. The line through the foci intersects the hyperbola at two points, the vertices.
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The line segment joining the vertices is the
transverse axis and its midpoint is the center
of the hyperbola. A hyperbola has two
branches and two asymptotes.
The asymptotes contain the diagonals of a rectangle centered at the hyperbola’s center, as shown
The standard form of the equation of a hyperbola with center at (0, 0) is as follows
Equation Transverse axis Vertex Foci Asymptotes
12
2
2
2
b
y
a
x Horizontal )0,( a )0,( c xy
ab
12
2
2
2
b
x
a
y Vertical ),0( a ),0( c xy
b
a
The foci of the hyperbola lie on the transverse axis, c units from the center where 𝑎2 + 𝑏2 = 𝑐2
Example 1 Draw the hyperbola given by 4𝑥2 − 9𝑦2 = 36
Solution
First rewrite the equation in standard form 149
22
yx
. From the equation 𝑎2 = 9 and 𝑏2 = 4,
so 𝑎 = 3 and 𝑏 = 2 . Because the 𝑥2-term is positive, the transverse axis is horizontal and the
vertices are at (-3, 0) and (3, 0). To draw the hyperbola, first draw a rectangle that is centered at
the origin, 2a= 6 units wide and 2b= 4 units high. Then show the asymptotes by drawing the
lines that pass through opposite corners of the rectangle. Finally, draw the hyperbola.
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Example 2 Write an equation of the hyperbola with foci at (0, -3) and (0, 3) and vertices at (0, -
2) and (0, 2).
Solution
The transverse axis is vertical because the
foci and vertices lie on the y-axis. The
center is the origin because the foci and the
vertices are equidistant from the origin.
Since the foci are each 3 units from the
center, c= 3. Similarly, because the vertices
are each 2 units from the center, 𝑎 = 2 You
can use these values of a and c to find b.
Now 𝑐2 = 𝑎2 + 𝑏2 thus 𝑏2 = 9 − 4 = 5
Because the transverse axis is vertical, the
standard form of the equation is
154
22
xy
Example 3 A hyperbolic mirror can be used to take panoramic photographs. A camera is
pointed toward the vertex of the mirror and is positioned so that the lens is at one focus of the
mirror. An equation for the cross section of the mirror is 1916
22
xy
where x and y are
measured in inches. How far from the mirror is the lens?
Solution
Notice from the equation that 162 a and 92 b , so using these values and the equation
222 bac to find the value of c now 5259162 cc
Since 𝑎 = 4 and 𝑐 = 5, the vertices are at (0, -4) and (0, 4) and the foci are at (0, -5) and (0, 5) .
The camera is below the mirror, so the lens is at (0, º5) and the vertex of the mirror is at (0, 4).
The distance between these points is 4 − (−5) = 9 The lens is 9 inches from the mirror.
Example 4 The diagram at the right shows the hyperbolic cross section of a sculpture located at
the Fermi National Accelerator Laboratory in Batavia, Illinois.
a) Write an equation that models the curved
sides of the sculptur
b) At a height of 5 feet, how wide is the
sculpture? (Each unit in the coordinate
plane represents 1 foot.)
Solution
a) From the diagram you can see that the
transverse axis is horizontal and 1a .
So the equation has the form 12
22
b
yx
Because the hyperbola passes through
the point (2, 13), we can substitute 2x
and 13y into the equation and solve
for b to get 𝑏 = 7.5.
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An equation of the hyperbola is 15.7 2
22
yx
b) At a height of 5 feet above the ground, 𝑦 = −8. To find the width of the sculpture, substitute
this value into the equation and solve for x. You get 𝑥 ≈ 1.46. At a height of 5 feet, the width
is 2𝑥 ≈ 2.92 feet.
Translated Hyperbola The standard equation of a translated hyperbola centered at point (h , k) and major and minor
axes of lengths 2a and 2b, where 𝑎 > 𝑏 > 0 is as follows
Equation Transverse axis Vertex Foci Asymptotes
1)()(
2
2
2
2
b
ky
a
hx Horizontal ),( kah
),( kch
. )(( hxky
a
b
1)()(
2
2
2
2
b
hx
a
ky Vertical ),( akh
),( ckh
)(( hxky
b
a
Example 1 Graph 14
)1()1(
22
xy
Solution The 2yy -term is positive, so the transverse axis is vertical. Since 4 and1 22 ba ,
Plot the center at (h, k) = (-1, -1). Plot the
vertices 1 unit above and below the center at
(-1, 0) and (-1, -2). Draw a rectangle that is
centered at (-1,-1) and is 2a = 2 units high
and 2b = 4 units wide. Draw the asymptotes
through the corners of the rectangle. Draw
the hyperbola so that it passes through the
vertices and approaches the asymptotes. If the origin is placed halfway between the foci, the equation is found using the distance formula.
Exercise
1) Graph the equation. Identify the vertices, the foci and asymptotes a) 99 22 xy
b) 13625
22
xy
c) 164
22
yx
d) 175100
22
xy
e) 3002512 22 xy
f) 18149
22
yx
g) 144436 22 yx h) 36418 22 yx i) 160208 22 xy
2) Write an equation of the hyperbola with the given foci and vertices )53,0(
c) Foci: (0, -5), (0, 5) Vertices: (0, -3), (0, 3)
d) Foci: (-8, 0), (8, 0) Vertices: (-7, 0), (7, 0)
e) Foci: )0,34( , )0,34( ) Vertices: (-5, 0), (5, 0)
f) Foci: (0, -9), (0, 9)Vertices: )53,0( , )53,0(
g) Foci: ( 8 , 0) , Vertices: )0,34( f) Foci: )65,0( , Vertices: (0, 4),)
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3) Suppose the mirror in example 3 above has a cross-section modeled by the equation
1925
22
yx
where x and y are measured in inches. If you place a camera with its lens at the
focus, how far is the lens from the vertex of the mirror?
4) Match the equation with its graph
a) 1416
22
yx
b) 124
22
xy
c) 1416
22
xy
d) 124
22
yx
5) Write the equation of the hyperbola in standard form a) 324936 22 yx b) 8181 22 xy
c) 9436 22 xy d) 093616 22 xy e) 436
22
xy f) 9
9
4
9
22
yx
6) Identify the vertices and foci of the hyperbola a) 1649
22
yx
b) 149
22
xy
c) 14121
22
yx
d) 324814 22 xy e) 100425 22 xy f) 3601036 22 yx
7) Graph the equation. Identify the foci and asymptotes a) 112125
22
yx
b) 136
22
yx
c) 14925
22
xy
d) 11009
22
xy
e) 164
22
xy
f) 18125
16 22
yx
g) 14
9
64
22
yx
h) 810081100 22 yx i) 259 22 yx
8) The sundial at the left was designed by
Professor John Shepherd. The shadow of
the gnomon traces a hyperbola
throughout the day. Aluminum rods
form the hyperbolas traced on the
summer solstice, June 21, and the winter
solstice, December 21.
a) One focus of the summer solstice
hyperbola is 207 inches above the
ground. The vertex of the aluminum
branch is 266 inches above the
ground. If the x-axis is 355 inches
above the ground and the center of
the hyperbola is at the origin, write an equation for the summer solstice hyperbola
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b) One focus of the winter solstice hyperbola is 419” above the ground. The vertex of the
aluminum branch is 387 inches above the ground and the center of the hyperbola is at the
origin. If the x-axis is 355 inches above the ground, write an equation for the winter
solstice hyperbola.
c) Use your equations from Exercises 64 and 65 to draw the lower branch of the summer
solstice hyperbola and the upper branch of the winter solstice hyperbola
9) When an airplane travels faster than the speed of sound, the sound waves form a cone behind
the airplane. If the airplane is flying parallel to the ground, the sound waves intersect the
ground in a hyperbola with the airplane directly above its center. A sonic boom is heard
along the hyperbola. If you hear a sonic boom that is audible along a hyperbola with the
equation 14100
22
yx
where x and y are measured in miles, what is the shortest horizontal
distance you could be to the airplane?
10) Suppose you are making a ring out of clay for a necklace. If you have a fixed volume of clay
and you want the ring to have a certain thickness, the area of the ring becomes fixed.
However, you can still vary the inner radius x and the outer radius y.
a) Suppose you want to make a ring with an area of 2 square inches. Write an equation
relating x and y.
b) Find three coordinate pairs (x, y) that satisfy the relationship from part (a). Then find the
width of the ring, y-x, for each coordinate pair.
c) How does the width of the ring, y-x, change as x and y both increase? Explain why this
makes sense.
11) Use the diagram at the right to show that
|d2 - d1| = 2a.
12) Two microphones, 1 mile apart, record
an explosion. Microphone A receives the
sound 2 seconds after Microphone B. Is
this enough information to decide where
the sound came from? Use the fact that
sound travels at 1100 feet per second
Classifying a Conic Sections from its Equation
The equation of any conic can be written in the form 𝐴𝑥2 + 𝐵𝑥𝑦 + 𝐶𝑦2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
The discriminant 𝐵2 − 4𝐴𝐶 of the equation can be used to determine which type of conic the
equation represents.
If 𝐵2 − 4𝐴𝐶 < 0 B = 0, and A = C the equation is a circle
If 𝐵2 − 4𝐴𝐶 < 0 and either B ≠ 0 or 𝐴 ≠ 𝐶 the equation is a ellipse
If 𝐵2 − 4𝐴𝐶 = 0 the equation is a parabola
If 𝐵2 − 4𝐴𝐶 > 0 the equation is a hyperbola
Note If B = 0, each axis of the conic is horizontal or vertical. If B ≠ 0, the axes are neither
horizontal nor vertical.
Example 1 Classify and graph the conic given by 2𝑥2 + 𝑦2 − 4𝑥 − 4 = 0.
Solution
Since𝐴 = 2, 𝐵 = 0 and 𝐶 = 1, the discriminant is 𝐵2 − 4𝐴𝐶 = 02 − 4 × 2 × 1 = −8
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Because 𝐵2 − 4𝐴𝐶 < 0 and A ≠ C, the graph is an ellipse To graph the ellipse,
First complete the square
163
)1(6)1(2
24)12(24)42(
2222
2222
yxyx
yxxyxx
Clearly center is (1 , 0) and 36 ba
3 c . A sketch of the ellipse is shown
Example 2 Classify and graph the conic
given by 05481443294 22 yxyx
Solution
Since A=4 B=0 and C=-9, the discriminant is 𝐵2 − 4𝐴𝐶 = 02 − 4 × 4 × −9 = 144 > 0
Thus the graph is a hyperbola. To graph the hyperbola, first complete the square
14
)8(
9
)4(36)8(9)4(4
57664548)6416(9)168(4548)1449()324(
2222
2222
yxyx
yxxyxx
Comparing this with 1)()(
2
2
2
2
b
ky
a
hx,
ℎ = −4, 𝑘 = −8, 𝑎 = 3, and 𝑏 = 2 To
draw the hyperbola, plot the center at
(ℎ, 𝑘) = (−4, −8) and the vertices at (-7, -
8) and (-1, -8). Draw a rectangle 2𝑎 = 6
units wide and 2𝑏 = 4 units high and
centered at (-4, -8). Draw the asymptotes
through the corners of the rectangle. Then
draw the hyperbola so that it passes through
the vertices and approaches the asymptotes.
Exercise
1) Write an equation for the conic section.
a) Circle with center at; i) (4, -1) and radius 7 ii) (9, 3) and radius 4 iii) (4, 2) and radius 3
b) Parabola with vertex at; i) (1, -2) and focus at (1, 1) ii) (-3, 1) and directrix 𝑥 = −8
c) Ellipse with vertices at (2, -3) and (2, 6) and foci at (2, 0) and (2, 3)
d) Ellipse with vertices at (-2, 2) and (4, 2) and co-vertices at (1, 1) and (1, 3)
e) Hyperbola with vertices at (5, -4) and (5, 4) and foci at (5, -6) and (5, 6)
f) Hyperbola with vertices at (-4, 2) and (1, 2) and foci at (-7, 2) and (4, 2)
g) Ellipse with foci at (2, -4) and (5, -4) and vertices at (-1, -4) and (8, -4)
h) Parabola with vertex at (3, -2) and focus at (3, -4)
i) Hyperbola with foci at (5, 2) and (5, -6) and vertices at (5, 0) and (5, -4)
2) Match the equation with its graph a) 036243649 22 yxyx b)
09422 xyy c) 036243649 22 yxyx d) 046422 yxxy e)
046422 yxyx f) 061541694 22 yxyx
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 122
3) Classify the conic section.
a) 04422 yxx
b) 02653 22 yxyx
c) 084722 yxyx
d) 01325 22 yxyx
e) 01021022 yxyx
f) 0302634 22 yxyx
g) 018394 22 yxyx
h) 036243649 22 yxyx
i) 0222251636 22 yxyx
j) 06041644 22 yxyx
k) 0625429 22 yxxy
l) 0820182516 22 yxyx
m) 09822 yxx
010482 2 xyx
n) 03740122012 22 yxyx
o) 05510549 22 yxyx
p) 042422 yxyx
q) 0233624169 22 yxxy
r) 06364216 22 yxxy
s) 0171642 yxx
4) Look back at Example *. Suppose there is a tower 25 miles east and 30 miles north of your
house with a range of 25 miles. Does the region covered by this tower overlap the regions
covered by the two towers in Example 5? Illustrate your answer with a graph.
5) Graph the equation. Identify the important characteristics of the graph, such as the center,
vertices, and foci.
a) 1)1()7( 22 yx
b) )2(3)4( 2 xy
c) 4)2()6( 22 yx
d) )3(12)7( 2 yx
e) 14
)3(
16
)8( 22
xy
f) 149
)6(
2
)3( 22
yx
g) 1916
)1( 22
yx
h) 1)4(16
22
yx
6) Classify the conic section and write its equation in standard form. Then graph the equation.
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 123
a) 044122 xyy
b) 0248622 yxyx
c) 01198729 22 yxyx
d) 0484484 22 yxyx
e) 01824 22 yxyx
f) 036241222 yxyx
g) 01281616 22 yyx
h) 07489 22 yxyx
i) 036121222 yxyx
j) 0942022 yxy
k) 012842 yxx
l) 0164163649 22 yxyx
7) A Gregorian telescope contains two mirrors whose cross sections can be modeled by the equations
0254,295729405 22 yx and 01440120 2 xy What types of mirrors are they?
8) The whisper dish shown at the left can be
seen at the Thronateeska Discovery Center
in Albany, Georgia. Two dishes are
positioned so that their vertices are 50 feet
apart. The focus of each dish is 3 feet from
its vertex. Write equations for the cross
sections of the dishes so that the vertex of
one dish is at the origin and the vertex of
the other dish is on the positive x-axis.
9) To practice making a figure eight, a figure
skater will skate along two circles etched
in the ice. Write equations for two
externally tangent circles that are each 6
feet in diameter so that the center of one
circle is at the origin and the center of the
other circle is on the positive y-axis.
10) When a pencil is sharpened the tip becomes a cone. On a pencil with flat sides, the
intersection of the cone with each flat side is a conic section. What type of conic is it?
11) A new crayon has a cone-shaped tip.
When it is used for the first time, a flat
spot is worn on the tip. The edge of the
flat spot is a conic section, as shown.
What type(s) of conic could it be?
12) Which of the following is an equation of the hyperbola with vertices at (3, 5) and (3, -1) and
foci at (3, 7) and (3, o3)? (a) 19
)2(
25
)3( 22
yx
(b) 125
)3(
9
)2( 22
xy
(c) 17
)3(
9
)2( 22
xy
(d) 116
)2(
9
)2( 22
yx
(e) 116
)3(
9
)2( 22
xy
13) What conic does 076210025 22 yxyx represent? (a) Parabola (b) Circle c) Ellipse
d) Hyperbola e) Not enough information
Time is precious, but we do not know yet how precious it really is. We will only know when we are no longer able to take advantage of it…
Proverbs 21:5 The plans of the diligent lead to profit as surely as haste leads to poverty By J. K. Kiingati Page 124
14) A degenerate conic occurs when the intersection of a plane with a double-napped cone is
something other than a parabola, circle, ellipse, or hyperbola.
a) Imagine a plane perpendicular to the axis of a double napped cone. As the plane passes
through the cone, the intersection is a circle whose radius decreases and then increases.
At what point is the intersection something other than a circle? What is the intersection?
b) Imagine a plane parallel to the axis of a double-napped cone. As the plane passes through
the cone, the intersection is a hyperbola whose vertices get closer together and then
farther apart. At what point is the intersection something other than a hyperbola? What is
the intersection?
c) Imagine a plane parallel to the nappe passing through a double-napped cone. As the plane
passes through the cone, the intersection is a parabola that gets narrower and then flips
and gets wider. At what point is the intersection something other than a parabola? What is
the intersection?
15) Tell what type of path each comet follows. Which comet(s) will pass by the sun more than
once?
a) 0000,137100200,14350 2 yxx
b) 0900,12400,18200,1346002200 22 yxyx
c) 0000,695000,52000,2026500000,5 2 yxyx