unit 1 quadratic functions & equations · 2019-08-30 · 1 unit 1 quadratic functions &...
TRANSCRIPT
1
Unit 1 Quadratic Functions &
Equations
General Outcome: • Develop algebraic and graphical reasoning through the study of relations.
Specific Outcomes:
1.1 Factor polynomial expressions of the form: determine the
• 2ax bx c+ + , 0a
• 2222 ybxa − , 0a , 0b
• ( ) ( ) cxfbxfa ++ )()(2
, 0a
• ( ) ( )2222 )()( ygbxfa − , 0a , 0b
where a, b, and c are rational numbers.
1.2 Analyze quadratic functions of the form ( ) qpxay +−=2
and determine the
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x and y-intercepts
1.3 Analyze quadratic functions of the form cbxaxy ++= 2 to identify characteristics of the
corresponding graph including:
• vertex
• domain and range
• direction of opening
• axis of symmetry
• x and y-intercepts
• and to solve problems.
1.4 Solve problems that involve quadratic equations.
2
Topics
• Graphing Quadratics Part I (Outcome 1.2) Page 3
• Graphing Quadratics Part II (Outcome 1.2) Page 8
• Completing the Square Part I (Outcome 1.3) Page 24
• Completing the Square Part II (Outcome 1.3) Page 32
• Maximum & Minimum Problems (Outcome 1.4) Page 37
• Solving Quadratic Equations by Graphing (Outcome 1.4) Page 43
• Solving Quadratic Equations by Factoring (Outcome 1.1 & 1.4) Page 49
• Solving Quadratic Equations by (Outcomes 1.3 & 1.4) Page 59
Completing the Square
• Solving Quadratic Equations using the (Outcome 1.4) Page 68
Quadratic Formula / the Discriminant
3
Unit 1 Quadratic Functions &
Equations
Graphing Quadratics Part I:
What is a Quadratic?
A quadratic is an expression of degree 2.
Ex)
Graph of 2y x= :
The most basic quadratic function is given by 2y x= .
Create a table of values for 2y x= and then graph it on the
grid provided.
x y
4−
3−
2−
1− 0 1 2 3 4
4
Graphing Quadratics of the form 2y x q= + :
Below is the graph of 2y x=
Sketch the graph of Sketch the graph of 2 3y x= + 2 7y x= −
Rule: Function Notation:
5
Graphing Quadratics of the form 2( )y x p= − :
Below is the graph of 2y x=
Sketch the graph of Sketch the graph of 2( 5)y x= + 2( 8)y x= −
Rule: Function Notation:
6
In general if 2( )y x p q= + + then
0p the graph moves __________
0p the graph moves __________
0q the graph moves __________
0q the graph moves __________
The vertex of the parabola is given by __________.
Ex) Sketch the following parabolas. Then answer the
following.
a) 2( 4) 5y x= + − b) 2( 7) 2y x= − +
Coordinates of Vertex: Coordinates of Vertex:
Axis of Symmetry: Axis of Symmetry:
Domain: Range: Domain: Range:
x-intercept(s): y-intercept: x-intercept(s): y-intercept:
7
Ex) Determine the equation of the following parabolas.
a) b)
Ex) Describe what happens to the point ( )4, 16− on the graph
of 2y x= when the graph of 2y x= is changed into the
graph of 2( 7) 18y x= + + .
8
Graphing Quadratics Part I Assignment:
1) Graph the following quadratics without the use of a calculator.
a) 2( 4) 7y x= − − b) 2( 6) 1y x= + +
c) 2( 8) 3y x= − + d) 2( 5) 6y x= + −
9
2) Determine the equation of each parabola.
a) b)
c) d)
10
3) Without graphing, determine each of the following.
a) 2( 3) 11y x= + − b) 2( 14) 26y x= − +
Coordinate of the Vertex: Coordinates of the Vertex:
Equation of the axis Equation of the axis
of Symmetry: of Symmetry:
Domain: Domain:
Range: Range:
4) Describe what happens to the point ( )3, 9 on the graph of 2y x= when the
graph of 2y x= is changed into the graph of
2( 13) 9y x= − − .
5) Match each graph shown below with its equation.
( )2
3 6y x= + +
( )2
6 3y x= + −
( )2
6 3y x= − −
( )2
3 6y x= − +
11
Graphing Quadratics Part II:
Graphing Quadratics of the form 2y ax= :
Below is the graph of 2y x=
Sketch the graph of Sketch the graph of 23y x= 23y x= −
12
Below is the graph of 2y x=
Sketch the graph of Sketch the graph of 21
2y x= 21
2y x−=
Rule: Function Notation:
13
In general if 2( )y a x p q= − + then
( ), p q gives _______________
if 0a then ________________
if 0a then ________________
a also determines ____________
Ex) Without using your calculator, graph the following
quadratics and determine its characteristics.
a) 22 5y x= −
Coordinates of the Vertex:
Equation of the axis of Symmetry:
Domain & Range:
Minimum or Maximum Value:
b) 21 ( 4) 2
3y x−= − +
Coordinates of the Vertex:
Equation of the axis of Symmetry:
Domain & Range:
Minimum or Maximum Value:
14
c) 23( 7) 8y x= − + +
Coordinates of the Vertex:
Equation of the axis of Symmetry:
Domain & Range:
Minimum or Maximum Value:
d) 22 ( 2)
5y x= −
Coordinates of the Vertex:
Equation of the axis of Symmetry:
Domain & Range:
Minimum or Maximum Value:
Ex) Determine the equation for the parabola with its vertex at
( )1, 4− that goes through the point ( )2, 2− .
15
Ex) Determine the equation of the parabola that goes through
the point ( )14, 22 and has its vertex at ( )11, 14− .
Ex) Determine the equation of the parabola that has an axis of
symmetry given by 17x = − , a range of y R where 5y ,
and goes through the point ( )24, 16− − .
Ex) Determine the equation of the parabola that passes through
( )8, 5 and whose x-intercepts are 2− and 6.
16
Ex) Describe the transformation required to turn the graph of 2y x= into the graph of 26( 7) 10y x= − + − .
Ex) A rocket is fired into the air. Its height is given by the
equation 2( ) 4.9( 5) 124h t t= − − + , where h is the height in
meters and t is the time in seconds.
a) Sketch the graph of this equation.
b) What is the maximum height reached by the rocket?
and when does it reach its maximum height?
c) What was the height of the rocket when it was fired?
17
Ex) The stainless steel Gateway Arch in St. Louis, Missouri,
has the shape of a catenary which is a curve that
approximates a parabola. If the curve is graphed on a grid
it can be modeled by the equation 2( ) 0.02 192h d d= − + ,
where d is the horizontal distance from the centre of the
arch and h is the height of the arch.
a) Sketch a graph of the shape of the arch.
b) Find the height of the arch
c) Find the approximate width of the arch at its base.
d) Find the approximate height of the arch at a horizontal
distance 15 m form one end.
18
Graphing Quadratics Part II Assignment:
1) Complete the table of values for the various transformations on the graph
of 2y x= .
x
( )f x
2 ( )f x
5 ( )f x 1
( )2
f x 3
( )4
f x
4− 16
3− 9
1− 1
0 0
1 1
2 4
5 25
2) Describe how the graph of 23y x= can be obtained from the graph of
2y x= .
3) Describe how the graph of 21
5y x= can be obtained from the graph of
2y x= .
19
4) Graph the following quadratics without the use of a calculator.
a) 22( 4) 8y x= − − b) 21( 3) 6
4y x
−= + +
c) 23( 7) 9y x= − − d) 22
( 2) 63
y x−
= + +
20
5) Without graphing, determine each of the following.
a) 2( 3) 21y x= − − + b) 20.8( 14) 8y x= + +
Coordinate of the Vertex: Coordinates of the Vertex:
Equation of the axis of Equation of the axis
of Symmetry: of Symmetry:
Domain: Domain:
Range: Range:
Maximum or Minimum Value: Maximum or Minimum Value:
c) 23( 14) 12y x= − − + d)
29 3
62 8
y x
= + −
Coordinate of the Vertex: Coordinates of the Vertex:
Equation of the axis of Equation of the axis
of Symmetry: of Symmetry:
Domain: Domain:
Range: Range:
Maximum or Minimum Value: Maximum or Minimum Value:
21
6) Determine the equation of each graph given below in vertex form.
a) b)
c) d)
22
7) Determine the equation of the parabola whose vertex it located at ( )0, 6− and
goes through the point ( )3, 21 .
8) Determine the equation of the parabola whose vertex it located at ( )3, 10− −
and goes through the point ( )2, 5− .
9) The point ( )4, 16 is on the graph of 2y x= . Describe what happens to this
point when the graph is stretched vertically by a factor of 5 and then translated
7 units to the left and 9 units up.
10) The point ( )3, 9 is on the graph of 2y x= . Describe what happens to this
point when the graph is stretched vertically by a factor of 7
3
− and then
translated 4 units to the right and 5 units down.
23
11) The finance team at an advertising company is using the quadratic function
( )2
( ) 2.5 36 20000N x x= − − + to predict the effectiveness of a TV commercial
for a certain product, where N is the predicted number of people who buy the
product if the commercial is aired x times per week.
a) Explain how you could sketch the graph of the function, and identify its
characteristics.
b) According to this model, what is the optimum number of times the
commercial should be aired?
c) What is the maximum number of people that this model predicts will buy
the product?
12) The main section of the suspension bridge in Parc de la Gorge de Coaticook,
Quebec, has cables in the shape of a parabola. Suppose that the points on the
tops of the towers where the cables are attached are 168 m apart and 24 m
vertically above the minimum height of the cables. Determine the equation that
describes the shape of the cables if the origin is located at the top of the left
tower.
24
Completing the Square Part I:
When quadratics are written in the form
2( )y a x p q= − +
it is easy to graph them, this is known as Standard Form.
Quadratics can also be written in General Form
2y Ax Bx C= + + , where A,B, and C are Integers
In order to graph parabolas in general form we must first change
it to standard form, this is done using a process called
completing the square.
Factor the following.
2 6 9x x+ + 2 10 25x x− + 2 22 121x x+ +
What do you notice about the coefficient of the middle term and
the constant?
Standard Form: 2( )y a x p q= − +
General Form: 2y Ax Bx C= + +
25
Ex) Find the value of c that makes each of the following a
perfect square trinomial.
a) 2 14x x c+ + b) 2 20x x c− +
c) 2 7x x c+ + d) 2 53
a a c− +
Ex) Convert the following to Standard Form.
a) 2 6y x x= + b) 2 8y x x= −
c) 2 10 30y x x= + + d) 2 2y x x= + −
26
e) 2 7 5y x x= + +
f) 2 12 11 0x y x− + − =
g) 23 3 7 2 0x y x− + + − =
27
Completing the Square Part I Assignment:
1) Find the value of c that will make each expression a perfect square trinomial.
a) 2 12x x c− + b) 2 18x x c+ +
c) 2 20x x c+ + d) 2 5x x c+ +
e) 2x x c− + f) 2 0.05x x c− +
g) 2 13.7x x c+ + h) 2
6
xx c+ +
2) Write each function in the form 2( )y a x p q= − + . Sketch the graph, showing
the coordinates of the vertex, the axis of symmetry, and the coordinates of two
other points on the graph.
a) 2 4 1y x x= − −
28
b) 2 2 3y x x= − +
c) 212 8y x x= − +
d) 2 17 10y x x= + +
29
3) The six graphs represent the six equations given below. Match each graph with
the correct equation.
a) 2 4y x x= + b) 2 4y x x= − c) 2 4y x x= − +
d) 2 4y x x= − − e) 2 4y x= − f) 2 4y x= − +
30
4) State the maximum or minimum value of y and the value of x when it occurs.
a) 2 3 3y x x= + + b) 2 2y x x= − −
5) Sketch the graph of the function given below and state any intercepts and the
coordinates of the vertex.
( 1)( 3)y x x= + +
31
6) A student is asked to do the following “Choose any number and square it.
Then, subtract eight times the original number. Then, add 35. Find the value of
the original number that gives the least result.”
a) If x is the original number and y is the result, write an equation that
represents the instructions.
b) Find the original number that gives the least result.
7) Write an equation in general form for the parabola that passes through the
points ( )1, 4− , ( )2, 5− , ( )3, 0 .
8) For the function 2 6y x x k= + + , what value(s) of k will result in
a) one x-intercept? b) two x-intercepts? c) no x-intercepts?
32
Completing the Square Part II:
So far we have only completed the square when the coefficient
in front of the 2x term is 1. If this value is something other than
1 we must first factor it out from the 2x and the x terms.
Ex) Express the following in standard form.
a) 23 18 32y x x= + +
b) 22 20 17y x x= − + +
c) 2 8 6y x x= − + −
33
d) 25 3 12y x x= + −
e) 22 1 43 5 7
y x x= − +
34
Completing the Square Part II Assignment:
1) Express the following in standard ( )2
y a x p q= − + (vertex form) by
completing the square.
a) 22 12y x x= − b) 2( ) 6 24 17f x x x= + +
c) 2( ) 10 160 80f x x x= − + d)
23 42 96y x x= + −
e) 2( ) 4 16f x x x= − + f)
220 400 243y x x= − − −
35
g) 2 42 96y x x= − + − h) 2( ) 7 182 70f x x x= + −
i) 2 3
72
y x x= + − j) 2 3( )
8g x x x= − −
k) 2 5( ) 2 1
6f x x x= − + l) 22 18
33 5
y x x= + −
36
2) The managers of a business are examining costs. It is more cost-effective for
them to produce more items. However, if too many items are produced, their
costs will rise because of factors such as storage and overstock. Suppose that
they model the cost, C, of producing n thousand items with the function 2( ) 75 1800 60000C n n n= − + . Determine the number of items produced that
will minimize their costs.
3) Identify and correct the error(s) in the questions given below.
a)
( )
( )
( )
2
2
2
2
( ) 2 9 55
( ) 2 4.5 20.25 20.25 55
( ) 2 4.5 20.25 40.5 55
( ) 2 4.5 95.5
f x x x
f x x x
f x x x
f x x
= − −
= − + − −
= − + − −
= − −
b)
( )
( )
( )
2
2
2
2
8 16 5
8 2 1 1 5
8 2 1 8 5
8 1 13
y x x
y x x
y x x
y x
= − + +
= − + + − +
= − + + + +
= − − +
37
Maximum & Minimum Problems:
Maximum and minimum word problems are ones that involve a
quadratic equation and require us to find either a maximum or
minimum situation.
Steps for Solving:
• Create a quadratic equation that describes the quality that
you are finding the maximum or minimum of.
• Express your equation from the above step in standard
form (complete the square).
• Determine the location of the vertex from the changed
equation. This will tell you the maximum or minimum
value as well as when it happens.
Ex) Find two numbers whose difference is 6 and whose
product is a minimum.
38
Ex) A farmer wants to make a corral along a river. If he has 84
meters of fencing and if the river will act as one side of the
corral, to what dimensions should the corral be built so the
area contained within is a maximum?
Ex) 3 pens are to be created as shown below. If 280 m of
fencing is available to create the pens, to what dimensions
should they be built so that the area is a maximum?
39
Ex) A hotel books comedians for a festival every year.
Currently they charge $28 per ticket and at this price sell
all 500 tickets. The hotel is thinking of raising the ticket
price. If they know that for every $4 increase in the price
of the ticket 20 fewer people will attend, what price should
be charged per ticket to create a maximum profit?
40
Maximum & Minimum Problems Assignment:
1) A concert promoter is planning the ticket price for an upcoming concert for a
certain band. At the last concert, she charged $70 per ticket and sold 2000
tickets. After conducting a survey, the promoter has determined that for every
$1 decrease in ticket price, she might expect to sell 50 more tickets. For how
much should each ticket be sold in order to maximize the revenue and how
many tickets can the promoter expect to sell?
2) A holding pen is being built alongside a long building. The pen requires only
three fenced sides, with the building forming the forth side. There is enough
material for 90 m of fencing. To what dimensions should the pen be built in
order to maximize the area contained within?
41
3) Determine the value of two numbers that have a sum of 29 and whose product
is a maximum.
4) A set of fenced-in areas, as shown below is being planned on an open field. A
total of 900 m of fencing is available. How should the enclosure be built in
order to maximize the area within?
x
x
y y y
42
5) Determine the value of two numbers that have a difference of 13 and whose
product is a minimum.
6) The manager of a bike store is setting the price for a new model. Based on past
sales history, he predicts that if he sets the price at $360, he can expect to sell
280 bikes this season. He also predicts that for every $10 increase in the price,
he expects to sell five fewer bikes. At what price should he sell each bike in
order to maximize his revenue?
43
Quadratic Functions vs. Quadratic Equations
Quadratic Functions Quadratic Equations
Are expressions relating Are equations that involve
two variables together. one variable that can be
Can be represented by solved for.
a graph.
Ex) 2 6 3y x x= + − Ex) 2 5 6x x+ = −
Solving Quadratic Equations by Graphing:
Steps for Solving
• Bring all terms to one side of the equation (let it = 0).
• Graph the related function.
• Find the x-intercepts.
To solve: 22 20 42 0x x+ + =
Consider: 22 20 42y x x= + +
when 0y =
This means:
Find the x-intercepts
of 22 20 42y x x= + +
44
Ex) Solve 22 20 42x x+ = −
Possible Outcomes when solving quadratic equations:
2 Real Roots 1 Real Root No Real Roots
Ex) Solve the following by graphing a related function on your
calculator. Round your answers to the nearest hundredth if
necessary.
a) 2 6x x+ = b) 22 3 34x x= +
45
c) 2 10 25 18x x+ + = d) 210 2 8 18x x− = − +
e) 2 19 8x x+ = − f) ( )2
0.1 3 1.6m+ =
Ex) The hypotenuse of a right triangle measures 10 cm. One
leg of the triangle is 2 cm longer than the other. Find the
lengths of the legs for the triangle.
46
Ex) The function 2( ) 0.025h d d d= − + models the behavior of
a soccer ball when it is kicked. h represents the height of
the ball in metres and d represents the horizontal distance
the ball has traveled in metres. Determine the horizontal
distance the ball will travel when it hits the ground.
47
Solving Quadratic Equations by Graphing Assignment:
1) Determine the roots of each equation by graphing a corresponding function.
Round answers to the nearest tenth if necessary.
a) 2 5 24 0x x− − = b) 2 2 5 0b b+ + = c) 2 4 4a a− + =
d) 23 9 12 0x x+ − = e) 2 4 3 0a a− + − = f) 2 10 21x x− = −
2) In a Canadian Football League game, the path of the football at one particular
kick-off can be modelled using the function 2( ) 0.02 2.6 66.5h d d d= − + − ,
where h is the height of the ball and d is the horizontal distance from the
kicking team’s goal line, both in yards. A value of ( ) 0h d = represents the
height of the ball at ground level. What horizontal distance does the ball travel
before it hits the ground?
3) Two numbers have a sum of 9 and a product of 20. Determine a single-variable
equation that can be used to determine the product and then determine the
values of the numbers by graphing a related function.
48
4) The path of the stream of water coming out of a fire hose can be approximated
using the function 2( ) 0.09 1.2h x x x= − + + , where h is the height of the water
stream and x is the horizontal distance from the firefighter holding the nozzle,
both in metres. At what maximum distance from the building could a
firefighter stand and still reach the base of the fire with water?
5) The height of a circular arch is represented by 2 24 8 0h hr s− + = , where h is
the height , r is the radius, and s is the span of the arch, all in feet. How high
must an arch be to have a span of 64 ft and a radius of 40 ft?
6) Suppose the value of a quadratic function is negative when 1x = and positive
when 2x = . Explain why it is reasonable to assume that the related equation
has a root between 1 and 2.
7) The equation on the axis of symmetry of a quadratic function is 3x = and one
x-intercept at 4− . Determine the location of the other x-intercept.
49
Solving Quadratic Equations by Factoring:
Steps for solving
• Bring all terms to one side of the equations (let it = 0).
• Factor the equation.
• Let each factor = 0 and solve.
Ex) Solve the following by factoring.
a) 2 12 35x x+ = − b) 23 13 10x x− =
c) 2 25 0x − = d) 22 3 5 1x x+ = +
If 0ab = then
0a = or 0b =
50
e) 23 5 0x x+ = f) 2 5
61 1x x+ = −
+ −
g) 2 4 21x x+ = h) 24 5 21 0x x− − =
i) 2 10 24 0x x+ + = j) 2 10 25x x+ = −
k) 26 5 2 7 2x x x− + = + l) 2 22 13a a+ = −
51
m) 2 64 0x − = n) 22 40 8x x− = − +
o) 25 52 20a a= − p) 2 16 64 0x x− + =
q) 2 15 11x − = − r) 26 20 0x x+ =
s) 24 20 25 0x x− + = t) 2 23 6 10 2 6x x x+ − = +
52
Ex) Determine a quadratic equation that has roots of 14
− and
5.
Ex) Solve the following quadratic equation.
(2 3) 4( 1) 2(3 2 )x x x x− + + = +
Ex) The length of a soccer pitch is 20 m less than twice its
width. The area of the pitch is 6000 m2. Find its
dimensions.
53
Solving Quadratic Equations by Factoring Assignment:
1) Completely factor each of the following.
a) 2 7 10x x+ + b) 25 40 60a a+ + c) 20.2 2.2 5.6x x− +
d) 2 24 9y x− e) 28 6 5b b− − f) 20.4 0.6 1.8x x+ −
g) 2 20x x+ − h) 21 9
4 25x − i)
212 3
4x x+ +
j) 22 12 18a a+ + k)
23 4 7y y+ − l) 2 12 36x x− +
54
m) ( ) ( )2
2 2 42x x+ − + − n) ( ) ( )2
2 26 4 4 4 4 1x x x x− + + − + −
o) ( ) ( )2 2
4 2 2 4x x− − + p) ( ) ( )2
4 5 3 10 5 3 6x x− + − −
q) ( ) ( )2 2216 1 4 2x x+ − r) ( ) ( )
22 312 25 2
4x y
−+
55
2) Determine the roots of each factored equation.
a) ( )( )3 5x x+ − b) ( )1
22
x x
− +
c) ( )9x x −
d) ( )( )3 1 5 4x x+ − e) ( )( )2 7 8 5x x− + f) ( )( )9 2 6x x− +
3) Solve the following quadratic equations by factoring.
a) 210 40 0x − = b)
2 42 0x x+ − =
c) 23 28 9 0a a+ + = d) 21 5
1 04 4
x x+ + =
e) 2 9 0y y− = f)
29 8x x= +
56
g) 2259 0
9x − = h) 22 4 70x x− =
i) 242 x x= − j) 23 9 30x x+ =
k) 22 3 5a a= − l) 2 4 21y y+ =
4) A rectangle has a length of ( )10x + and a width of ( )2 3x − . If the area of the
rectangle is 54 cm2 determine its length and width.
57
5) An osprey, a fish-eating bird of prey, dives toward the water to catch a salmon.
The height, h, in metres, of the osprey above the water t seconds after it begins
its dive can be approximated by the function 2( ) 5 30 45h t t t= − + . Determine
the time it takes for the osprey to reach a height of 20 m.
6) The area of a square is tripled by adding 10 cm to one dimension and 12 cm to
the other. Determine the side length of the original square.
7) An 18 m tall tree is broken during a severe storm, as shown, The distance from
the base of the trunk to the point where the tip touches the ground is 12 m. At
what height did the tree break?
58
8) A rectangle with area of 35 cm2 is formed by cutting off strips of equal width
from a rectangular piece of paper. Determine the width of each strip.
59
Solving Quadratic Equations by Completing the Square:
Ex) Solve the following:
2 81x = ( )
23 100x − =
Expressions of the Form 02 =+ cax or 0)( 2 =+− cpxa Can
be Solved by:
• Isolate the term that is squared.
• Take the square root of both sides of the equation
*Remember when taking the square root we must consider
both the positive and negative situation
• Break equation into 2 separate equations (+’ve case and
–‘ve case), then solve each separately.
*Leave answers in reduced radical form if necessary.
Ex) Solve the following using the square root principle.
a) 2( 3) 16x + = b) 29 21 0x − =
60
c) 2( 5) 6 30a + − = d) 23( 2) 84x − =
e) 22(2 6) 88x − = f) 24( 1) 3 10y + + =
Equations in the Form 2 0ax bx c+ + = Can be solved by:
• Bring all terms to one side of the equation (let it = 0)
• Complete the square putting the one side of the equation
into standard form.
• Isolate the term that is squared.
• Take the square root of both sides of the equation
*Remember when taking the square root we must consider
both the positive and negative situation
• Break equation into 2 separate equations (+’ve case and
–‘ve case), then solve each separately.
*Leave answers in reduced radical form if necessary.
61
Ex) Solve the following by completing the square.
a) 2 6 27 0x x+ − = b) 22 10x x− =
c) 2 8 11 0x x+ + = d) 22 5 1x x− =
62
e) 24 12 9 0x x− + = f)
2 8 21x x− = −
g) 2 14 1 0x x+ + = h) 23 11 10x x− =
63
Ex) A picture that measures 10 cm by 5 cm is to be surrounded
by a mat before being framed. The width of the mat is to
be the same on all sides of the picture. The area of the mat
is to be twice the area of the picture. What is the width of
the mat?
64
Solving Quadratic Equations by Completing the Square Assignment:
1) Solve the following quadratic equations using the square root principle.
a) ( )2
3 4x − = b) ( )2
2 9x + =
c)
21
12
d
+ =
d)
23 7
4 16a
− =
e) ( )2 3
64
x + = f) ( )2
4 18x + =
65
2) Solve the following quadratic equations by completing the square.
a) 2 10 4 0x x+ + = b) 2 8 13 0x x− + =
c) 23 6 1 0x x+ + = d) 22 4 3 0x x− + + =
e) 20.1 0.6 0.4 0x x− − + = f)
2 8 4 0x x− − =
66
g) 23 4 5 0x x− + + = h) 216 5 0
2x x− − =
i) 20.2 0.12 11 0x x+ − = j) 222 0
3x x
−− + =
3) Doug’s rectangular dog kennel measures 4 ft by 10 ft. He plans to double the
area of the kennel by extending each side by an equal amount. Write an
equation to model the new area and determine the new dimensions of the
kennel to the nearest tenth of a ft.
67
4) Write a quadratic equation with roots of 7 and 7− .
5) Write a quadratic equation with roots of ( )1 3+ and ( )1 3− .
68
Solving Quadratic Equations using the Quadratic Formula:
Complete the square and then solve the following:
2 0ax bx c+ + =
Ex) Solve the following using the quadratic formula.
a) 2 3 28 0x x+ − = b) 23 5 2x x+ =
c) 2 2 1 0x x− − = d) 2
2
2 3 2
1 2 2
x
x x x x+ =
+ − − −
Quadratic Formula:
2 4
2
b b acx
a
− −=
69
Ex) Lindsay travelled from Calgary to Spokane, a distance of
720 km. On the return trip her average speed was 10 km/h
faster. If the total driving time was 17 hours, what was
Lindsay’s average speed from Calgary to Spokane?
70
The Discriminant:
The Discriminant is the part of the quadratic formula that is
inside the square root.
2 4
2
b b acx
a
− −= 2 4b ac−
• If 2 4 0b ac− then,
• If 2 4 0b ac− = then,
• If 2 4 0b ac− then,
Ex) Determine the nature of the roots for the following.
(Determine how many answers the following will have.)
a) 2 5 0x x− − = b) 2 8 16 0x x− + =
71
c) 23 11 12 0x x− + = d) 22 3 4x x+ =
Ex) Find the value of k so that the following equation has no
real roots.
23 22 0x x k− + =
72
Solving Quadratic Equations Using the Quadratic Formula / The
Discriminant Assignment:
1) Determine the nature of the roots (number of solutions) for each equation given
below.
a) 2 7 4 0x x− + = b) 2 9 6 0b b+ + = c) 2 2 1 0a a− + =
d) 27 3 2 0x x+ + = e)
2 2 14x x= + f) 23 0.06 4 0x x− + + =
g) 213 9 0
4b b− + = h)
2 2 1 0x x− + − = i) 21 5
2 2x x+ =
73
2) Solve the following quadratic equations using the quadratic formula.
a) 27 24 9 0x x+ + = b) 24 12 9 0a a− − =
c) 23 5 1x x+ = d) 22 4 7 0b b+ − =
e) 23 14 5 0x x+ + = f)
25 16 2 0a a− + − =
74
g) 28 12 1y y+ = − h) 210 45 7 0x x− − =
i) 23 6 1 0x x+ + = j) 2 10
6 2
bb + − =
k) 24 7 12 0x y+ − = l)
22 6 1x x= −
75
3) An open-topped box is being made from a piece of cardboard measuring 12 in.
by 30 in. The sides of the box are formed when four congruent squares are cut
from the corners, as shown in the diagram. The base of the box has an area of
208 sq. in. Determine the side length, x, of the square cut from each corner.
4) A study of the air quality in a particular city suggests that t years from now, the
level of carbon monoxide in the air, A, in parts per million, can be modelled by
the function 2( ) 0.3 0.1 4.2A t t t= + + . In how many years from now will the
carbon monoxide level be 8 parts per million. Round your answer to the
nearest tenth of a year.
76
Answers Graphing Quadratics Part I Assignment:
1. a) b)
c) d)
2. a) ( )2
4 8y x= − − b) ( )2
5 2y x= + + c) ( )2
7 7y x= + −
d) ( )2
5y x= −
3. a) Vertex: ( )3, 11− − Axis of Symmetry: 3x = − Domain: x x R
Range: 11,y y y R −
b) Vertex: ( )14, 26 Axis of Symmetry: 14x = Domain: x x R
Range: 26,y y y R
4. The point ( )3, 9 is translated 13 units to the right and 9 units down so that it is
becomes ( )16, 0 .
77
5. a) ( )2
3 6y x= + + b) ( )2
3 6y x= − + c) ( )2
6 3y x= + −
d) ( )2
6 3y x= − −
Graphing Quadratics Part II Assignment:
1.
x
( )f x
2 ( )f x
5 ( )f x 1
( )2
f x 3
( )4
f x
4− 16 32 80 8 12
3−
9
18
45 9
2
27
4
1−
1
2
5 1
2
3
4
0 0 0 0 0 0
1
1
2
5 1
2
3
4
2 4 8 20 2 3
5
25
50
125 25
2
75
4
2. The graph of 23y x= can be obtained from the graph of
2y x= by vertically
stretching it by a factor of 3
3. The graph of 21
5y x= can be obtained from the graph of
2y x= by vertically
stretching it by a factor of 1
5.
4. a)
b)
\
78
c)
d)
5. a) Vertex: ( )3, 21 Axis of Symmetry: 3x = Domain: x x R
Range: 21,y y y R Maximum of 21
b) Vertex: ( )14, 8− Axis of Symmetry: 14x = − Domain: x x R
Range: 8,y y y R Minimum of 8.
c) Vertex: ( )14, 12 Axis of Symmetry: 14x = Domain: x x R
Range: 12,y y y R Maximum of 12
d) Vertex: 3
, 68
− −
Axis of Symmetry:
3
8x
−= Domain: x x R
Range: 6,y y y R − Minimum of 6−
6. a) ( )2
3 4y x= + − b) ( )2
2 1 12y x= − − + c) ( )21
3 12
y x= − +
d) ( )21
3 44
y x−
= + +
7. 23 6y x= − 8. ( )
213 10
5y x= + −
9. The point ( )4, 16 will be stretched vertically to become ( )4, 80 and then
translated to end up at ( )3, 89−
10. The point ( )3, 9 will be stretched vertically to become ( )3, 21− and then
translated to end up at ( )7, 26−
11. a) This will create a parabola with a vertex at ( )36, 20000 that opens down
and has been vertically stretched by a factor of 2.5. It has an axis of symmetry
at 36x = and has a maximum of 20000. b) 36 times c) 20000 people
12. ( )21
84 24294
y x= − −
79
Completing the Square Part I Assignment:
1. a) 36 b) 81 c) 100 d) 25
4 e)
1
4 f) 0.000625 g) 46.9225
h) 1
144
2. a) ( )2
2 5y x= − −
b) ( )2
1 2y x= − +
c) ( )
24 4y x= − −
d) ( )2
5 8y x= + −
80
3. a)
b)
c)
d)
e)
f)
4) a) A minimum of 3
4 occurs when
3
2x
−= .
b) A minimum of 9
4
− occurs when
1
2x = .
81
5.
Vertex: ( )2, 1− −
x-intercepts: ( )3, 0− and ( )1, 0−
y-intercept: ( )0, 3
6. a) 2 8 35y x x= − + b) 4
7. 2 2 3y x x= − −
8. a) 9k = b) 9k c) 9k
Completing the Square Part II Assignment:
1. a) ( )2
2 3 18y x= − − b) ( )2
( ) 6 2 7f x x= + −
c) ( )2
( ) 10 8 560f x x= − − d) ( )2
3 7 243y x= + −
e) ( )2
( ) 4 2 16f x x= − − + f) ( )2
20 10 1757y x= − + +
g) ( )2
21 345y x= − − + h) ( )2
( ) 7 13 1253f x x= + +
i)
23 121
4 16y x
= + −
j)
23 9
( )16 256
g x x
= − + +
k)
25 263
( ) 224 288
f x x
= − +
l)
22 27 393
3 10 50y x
= + −
2. They should produce 12 000 items.
3. a) 2
2
( ) 2 9 55
( ) 2 4.5 20.25 20.25
f x x x
f x x x
= − −
= − + −( )( )( )( )
2
2
2
55
( ) 2 4.5 5.0625 5.0625 55
( ) 2 4.5 5.0625 10.125 55
( ) 2 2.25 65.125
f x x x
f x x x
f x x
−
= − + − −
= − + − −
= − −
b) 2
2
8 16 5
8
y x x
y x
= − + +
= − +( )( )( )( )
2
2
2
2 1 1 5
8 2 1 1 5
8 2 1 8 5
8 1 13
x
y x x
y x x
y x
+ − +
= − − + − +
= − − + + +
= − − +
82
Maximum & Minimum Problems Assignment:
1. Each ticket should be sold for $55 and they can expect to sell 1250 tickets.
2. 22.5 m by 45 m
3. 14.5 and 14.5
4. 75x = m 50y = m
5. 6.5− and 6.5
6. $460
Solving Quadratic Equations by Graphing Assignment:
1. a) 3x = − , 8x = b) no real roots c) 2a = d) 4x = − , 1x =
e) 1a = , 3a = f) 3x = , 7x =
2. 60 yards
3. 29 20x x− = , the numbers are 4 and 5
4. 12.2 m
5. 64 ft
6. If the value is negative when 1x = , then the graph will be located below the
x-axis. If the value is positive when 2x = , then the graph will be located above
the x-axis. Therefore the graph must cross the x-axis between 1x = and 2x = .
At this value the equation will have a root.
7. ( )10, 0
Solving Quadratic Equations by Factoring Assignment:
1. a) ( )( )5 2x x+ + b) ( )( )5 6 2a a+ + c) ( )( )0.2 4 7x x− −
d) ( )( )2 3 2 3y x y x− + e) ( )( )4 5 2 1b b− + f) ( )( )0.2 2 3 3x x− +
g) ( )( )5 4x x+ − h) 1 3 1 3
2 5 2 5x x
− +
i) ( )( )
16 2
4x x+ +
j) ( )( )2 3 3a a+ + k) ( )( )3 7 1y y+ − l) ( )( )6 6x x− −
m) ( )( )5 8x x− + n) ( )( )2 23 12 11 2 8 9x x x x− + − + o) 32x−
p) ( )10 10 7x x − q) ( )( )2 216 1 1x x x x+ + − + r) ( )( )6 610 10y x y x+ −
2. a) 3x = − and 5x = b) 1
2x
−= and 2x = c) 0x = and 9x =
d) 1
3x
−= and
4
5x = e)
5
8x
−= and 7x = f) 6x = − and
9
2x =
83
3. a) 2x = − and 2x = b) 7x = − and 6x = c) 9a = − and 1
3a
−=
d) 4x = − and 1x = − e) 0y = and 9y = f) 8
9x
−= and 1x =
g) 9
5x
−= and
9
5x = h) 5x = − and 7x = i) 6x = − and 7x =
j) 5x = − and 2x = k) 3a = − and 1
2a = l) 7y = − and 3y =
4. length = 13.5 cm, width = 4 cm
5. 1t = seconds and again at 5t = seconds
6. 15 cm
7. 5 m
8. 1 cm
Solving Quadratic Equations by Completing the Square Assignment:
1. a) 1x = and 5x = b) 5x = − and 1x = c) 3
2d
−= and
1
2d =
d) 3 7
4a
= e)
12 3
2x
− = f) 4 3 2x = −
2. a) 5 21x = − b) 4 3x = c) 3 6
3x
− = d)
2 10
2x
=
e) 3 13x = − f) 4 2 5x = g) 2 19
3x
= h) 6 46x =
i) 3 5509
10x
− = j)
3 57
4x
− =
3. 24 28 40 0x x+ − = Dimensions: 12.4 ft by 6.4 ft
4. 2 7 0x − =
5. 2 2 2 0x x− − =
Solving Quadratic Equations Using the Quadratic Formula / The
Discriminant Assignment:
1. a) 2 roots b) 2 roots c) 1 root d) no real roots e) 2 roots
f) 2 roots g) 1 root h) 1 root i) no real roots
2. a) 3x = − and 3
7x
−= b)
3 3 2
2a
= c)
5 37
6x
− =
84
d) 2 3 2
2b
− = e)
7 34
3x
− = f)
8 3 6
5a
=
g) 3 7
4y
− = h)
45 2305
20x
= i)
3 6
3x
− =
j) 1 73
12b
− = k)
3 2
2x
= l)
3 7
2x
=
3. 2 in.
4. 3.4 years