symmetry and coordinate graphs - vista peak prep...
TRANSCRIPT
© Glencoe/McGraw-Hill 87 Advanced Mathematical Concepts
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
3-1
Symmetry and Coordinate GraphsOne type of symmetry a graph may have is point symmetry. Acommon point of symmetry is the origin. Another type ofsymmetry is line symmetry. Some common lines of symmetryare the x-axis, the y-axis, and the lines y 5 x and y 5 2x.
point x-axis y-axis y 5 x y 5 2x
Example 1 Determine whether ƒ(x) 5 x3 is symmetric withrespect to the origin.
If ƒ(2x) 5 2ƒ(x), the graph has point symmetry.
Find ƒ(2x). Find 2ƒ(x).ƒ(2x) 5 (2x)3 2ƒ(x) 5 2x3
ƒ(2x) 5 2x3
The graph of ƒ(x) 5 x3 is symmetric with respectto the origin because ƒ(2x) 5 2ƒ(x).
Example 2 Determine whether the graph of x21 2 5 y2 is
symmetric with respect to the x-axis, the y-axis,the line y 5 x, the line y 5 2x, or none of these.
Substituting (a, b) into the equation yields a2 1 2 5 b2. Check to see if each test produces an equation equivalent to a2 1 2 5 b2.
x-axis a2 1 2 5 (2b)2 Substitute (a, 2b) into the equation.a2 1 2 5 b2 Equivalent to a2
1 2 5 b2
y-axis (2a)2 1 2 5 b2 Substitute (2a, b) into the equation.a2 1 2 5 b2 Equivalent to a2
1 2 5 b2
y 5 x (b)2 1 2 5 (a)2 Substitute (b, a) into the equation.a2 2 2 5 b2 Not equivalent to a2
1 2 5 b2
y 5 2x (2b)2 1 2 5 (2a)2 Substitute (2b, 2a) into the equation.b2 1 2 5 a2 Simplify.a2 2 2 5 b2 Not equivalent to a2
1 2 5 b2
Therefore, the graph of x2 1 2 5 y2 is symmetric with respect to the x-axis and the y-axis.
© Glencoe/McGraw-Hill 88 Advanced Mathematical Concepts
Symmetry and Coordinate Graphs
Determine whether the graph of each function is symmetric with respect
to the origin.
1. ƒ(x) 5 }2
x12} 2. ƒ(x) 5 x5 2 2
3. ƒ(x) 5 x3 2 4x 4. ƒ(x) 5 }3
x2
2
x}
Determine whether the graph of each equation is symmetric with respect to
the x-axis, the y-axis, the line y 5 x, the line y 5 2x, or none of these.
5. x 1 y 5 6 6. x2 1 y 5 2
7. xy 5 3 8. x3 1 y2 5 4
9. y 5 4x 10. y 5 x2 2 1
11. Is ƒ(x) 5 uxu an even function, an odd function, or neither?
Refer to the graph at the right for Exercises 12 and 13.
12. Complete the graph so that it is the graph of an odd function.
13. Complete the graph so that it is the graph of an even function.
14. Geometry Cameron told her friend Juanita that the graph of u yu 5 6 2 u3xu has the shape of a geometric f igure. Determine whether the graph of u yu 5 6 2 u3xu is symmetric with respect to the x-axis, the y-axis, both, or neither. Then make a sketch of the graph.Is Cameron correct?
Practice
NAME _____________________________ DATE _______________ PERIOD ________
3-1
x
f x( )
O
(3, 3)
(4, 2)(1, 1)
© Glencoe/McGraw-Hill 89 Advanced Mathematical Concepts
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
3-1
Symmetry in Three-Dimensional FiguresA solid figure that can be superimposed, point forpoint, on its mirror image has a plane of symmetry.A symmetrical solid object may have a finite orinfinite number of planes of symmetry. The chair inthe illustration at the right has just one plane ofsymmetry; the doughnut has infinitely manyplanes of symmetry, three of which are shown.
Determine the number of planes of symmetry for each object anddescribe the planes.
1. a brick
2. a tennis ball
3. a soup can
4. a square pyramid
5. a cube
Solid figures can also have rotational symmetry. For example, theaxis drawn through the cube in the illustration is a fourfold axis ofsymmetry because the cube can be rotated about this axis into fourdifferent positions that are exactly alike.
6. How many four-fold axes of symmetry does a cube have?
Use a die to help you locate them.
7. A cube has 6 two-fold axes of symmetry. In the space at the right,draw one of these axes.
© Glencoe/McGraw-Hill 90 Advanced Mathematical Concepts
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
3-2
Families of GraphsA parent graph is a basic graph that is transformed to create othermembers in a family of graphs. The transformed graph may appear ina different location, but it will resemble the parent graph.
A reflection flips a graph over a line called the axis of symmetry.A translation moves a graph vertically or horizontally.A dilation expands or compresses a graph vertically orhorizontally.
Example 1 Describe how the graphs of ƒ(x) 5 Ïxw andg(x) 5 Ï2wxw 2 1 are related.
The graph of g(x) is a reflection of the graph ofƒ(x) over the y-axis and then translated down 1 unit.
Example 2 Use the graph of the given parent functionto sketch the graph of each related function.
a. ƒ(x) 5 x3; y 5 x3 1 2
When 2 is added to the parent function, the graph of the parent function moves up 2 units.
b. ƒ(x) 5 vxb; y 5 3vxb
The parent function is expanded vertically by afactor of 3, so the vertical distance between thesteps is 3 units.
c. ƒ(x) 5 x; y 5 0.5x
When x is multiplied by a constant greaterthan 0 but less than 1, the graph compressesvertically, in this case, by a factor of 0.5.
d. ƒ(x) 5 x2; y 5 x2 2 4
The parent function is translated down 4 unitsand then any portion of the graph below the x-axis is reflected so that it is above the x-axis.
© Glencoe/McGraw-Hill 91 Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Families of Graphs
Describe how the graphs of ƒ(x) and g(x) are related.
1. ƒ(x) 5 x2 and g(x) 5 (x 1 3)2 2 1 2. ƒ(x) 5 x and g(x) 5 22x
Use the graph of the given parent function to describe the graph of each
related function.
3. ƒ(x) 5 x3 4. ƒ(x) 5 Ïxwa. y 5 2x3 a. y 5 Ïxw 1w 3w 1 1
b. y 5 20.5(x 2 2)3 b. y 5 Ï2wxw 2 2
c. y 5 (x 1 1)3 c. y 5 Ï0w.2w5wxw 2 4
Sketch the graph of each function.
5. ƒ(x) 5 2(x 2 1)2 1 1 6. ƒ(x) 5 2x 1 2 2 3
7. Consumer Costs During her free time, Jill baby-sits the neighborhood children. She charges $4.50 for each whole hour or any fraction of an hour. Write and graph a function that shows the cost of x hours of baby-sitting.
3-2
© Glencoe/McGraw-Hill 92 Advanced Mathematical Concepts
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
3-2
A graph G′ is isomorphic to a graph G if the following conditions hold.
1. G and G′ have the same number of vertices and edges.
2. The degree of each vertex in G is the same as the degree of each corresponding vertex in G′.3. If two vertices in G are joined by k (k $ 0) edges, then the two corresponding vertices in G′ are
also joined by k edges.
Isomorphic GraphsA graph G is a collection of points in which a pair of points, called vertices, are connected by a set of segments or arcs, called edges. The degree of vertex C, denoteddeg (C ), is the number of edges connected to that vertex. We say two graphs are isomorphic if they have the same structure. The definition below will help youdetermine whether two graphs are isomorphic.
Example In the graphs below HIJKLMN ↔ TUVWXYZ.Determine whether the graphs are isomorphic.
Number of vertices in G: 7 Number of vertices in G′: 7Number of edges in G: 10 Number of edges in G′: 10deg (H): 3 deg (I): 3 deg (T): 1 deg (U): 3deg (J): 3 deg (K): 3 deg (V): 3 deg (W): 3deg (L): 4 deg (M): 3 deg (X): 4 deg (Y): 3deg (N): 1 deg (Z): 3
Since there are the same number of vertices and the same number of edgesand there are five vertices of degree 3, one vertex of degree 4, and onevertex of degree 1 in both graphs, we can assume they are isomorphic.
Each graph in Row A is isomorphic to one graph in Row B. Match the graphs that are isomorphic.
Row A 1. 2. 3.
Row B a. b. c.
© Glencoe/McGraw-Hill 93 Advanced Mathematical Concepts
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
Graphs of Nonlinear InequalitiesGraphing an inequality in two variables identifies all orderedpairs that satisfy the inequality. The first step in graphingnonlinear inequalities is graphing the boundary.
Example 1 Graph y , Ïxw 2w 3w 1 2.
The boundary of the inequality is the graph of y 5Ïxw 2w 3w 1 2.To graph the boundary curve, start with the parent graph y 5Ïxw. Analyze the boundary equation to determine how theboundary relates to the parent graph.
y 5 Ïxw 2w 3w 1 2 ↑ ↑
move 3 units right move 2 units up
Since the boundary is not included in the inequality, thegraph is drawn as a dashed curve.
The inequality states that the y-values of the solution areless than the y-values on the graph of y 5Ïxw 2w 3w 1 2.Therefore, for a particular value of x, all of the points in theplane that lie below the curve have y-values less than Ïxw 2w 3w 1 2. This portion of the graph should be shaded.
To verify numerically, test a point not on the boundary.
y , Ïxw 2w 3w 1 20
?, Ï4w 2w 3w 1 2 Replace (x, y) with (4, 0).
0 , 3 3 True
Since (4, 0) satisfies the inequality, the correct region isshaded.
Example 2 Solve x 2 3 2 2 . 7.
Two cases must be solved. In one case, x 2 3 is negative, andin the other, x 2 3 is positive.
Case 1 If a , 0, then a 5 2a. Case 2 If a . 0, then a 5 a.2(x 2 3) 2 2 . 7 x 2 3 2 2 . 7
2x 1 3 2 2 . 7 x 2 5 . 72x . 6 x . 12
x , 26
The solution set is {xx , 26 or x . 12}.
3-3
© Glencoe/McGraw-Hill 94 Advanced Mathematical Concepts
Graphs of Nonlinear Inequalities
Determine whether the ordered pair is a solution for the given inequality.
Write yes or no.
1. y . (x 1 2)2 1 3, (22, 6) 2. y , (x 2 3)3 1 2, (4, 5) 3. y # u 2x 2 4 u 2 1, (24, 1)
Graph each inequality.
4. y # 2 u x 2 1 u 5. y . 2(x 2 1)2
6. y , Ïxw 2w 2w 1 1 7. y $ (x 1 3)3
Solve each inequality.
8. u 4x 2 10 u # 6 9. u x 1 5 u 1 2 . 6 10. u 2x 2 2 u 2 1 , 7
11. Measurement Instructions for building a birdhouse warn that the platform, which ideally measures 14.75 cm2, should not vary in size by more than 0.30 cm2. If it does, the preconstructed roof for the birdhouse will not f it properly.a. Write an absolute value inequality that represents the range of
possible sizes for the platform. Then solve for x to f ind the range.
b. Dena cut a board 14.42 cm2. Does the platform that Dena cut f it within the acceptable range?
Practice
NAME _____________________________ DATE _______________ PERIOD ________
3-3
© Glencoe/McGraw-Hill 95 Advanced Mathematical Concepts
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
Some Parametric GraphsFor some curves, the coordinates x and y can be written as functions of a third variable. The conditions determining the curve are given by two equations, rather than by a single equation in x and y. The thirdvariable is called a parameter, and the two equations are called para-metric equations of the curve.
For the curves you will graph on this page, the parameter is t and the parametric equations of each curve are in the form x 5 f (t) and y 5 g (t).
Example Graph the curve associated with the parametricequations x 5 48t and y 5 64t 2 16t2.
Choose values for t and make a table showing the values of all three variables. Then graph thex- and y-values.
t x y
–1 –48 –800 0 00.5 24 281 48 482 96 643 144 484 192 0
Graph each curve.
1. x 5 3t, y 5 2. x 5 t2 1 1, y 5 t3 2 112}t
3-3
© Glencoe/McGraw-Hill 96 Advanced Mathematical Concepts
Inverse Functions and Relations
Two relations are inverse relations if and only if one relationcontains the element (b, a) whenever the other relationcontains the element (a, b). If the inverse of the function ƒ(x)is also a function, then the inverse is denoted by ƒ21(x).
Example 1 Graph ƒ(x) 5 }14
}x32 3 and its inverse.
To graph the function, let y 5 ƒ(x). To graphƒ21(x), interchange the x- and y-coordinates ofthe ordered pairs of the function.
You can use the horizontal line test to determine if theinverse of a relation will be a function. If every horizontal lineintersects the graph of the relation in at most one point, thenthe inverse of the relation is a function.
You can find the inverse of a relation algebraically. First, lety 5 ƒ(x). Then interchange x and y. Finally, solve the resulting equation for y.
Example 2 Determine if the inverse of ƒ(x) 5 (x 2 1)21 2
is a function. Then find the inverse.
Since the line y 5 4 intersects the graph of ƒ(x) at more than one point, the function fails thehorizontal line test. Thus, the inverse of ƒ(x) is not a function.
y 5 (x 2 1)2 1 2 Let y 5 ƒ(x).x 5 ( y 2 1)2 1 2 Interchange x and y.
x 2 2 5 ( y 2 1)2 Isolate the expression containing y.
6Ïxw 2w 2w 5 y 2 1 Take the square root of each side.
y 5 1 6 Ïxw 2w 2w Solve for y.
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
3-4
ƒ (x) 5 }14
}x32 3
x y
23 29.75
22 25
21 23.25
0 23
1 22.75
2 21
3 3.75
ƒ21(x)
x y
29.75 23
25 22
23.25 21
23 0
22.75 1
21 2
3.75 3
© Glencoe/McGraw-Hill 97 Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Inverse Functions and Relations
Graph each function and its inverse.
1. ƒ(x) 5 (x 2 1)3 1 1 2. ƒ(x) 5 3ux u 1 2
Find the inverse of ƒ(x). Then state whether the inverse is also a function.
3. ƒ(x) 5 24x2 1 1 4. ƒ(x) 5 Ï3
xw 2w 1w 5. ƒ(x) 5 }(x 2
4
3)2}
Graph each equation using the graph of the given parent function.
6. y 5 2Ïxw 1w 3w 2 1, p(x) 5 x2 7. y 5 2 1 Ï5
xw 1w 2w, p(x) 5 x5
8. Fire Fighting Airplanes are often used to drop water on forest f ires in an effort tostop the spread of the fire. The time t it takes the water to travel from height h tothe ground can be derived from the equation h 5 }
12
}gt2 where g is the accelerationdue to gravity (32 feet/second2).
a. Write an equation that will give time as a function of height.
b. Suppose a plane drops water from a height of 1024 feet. How many seconds will ittake for the water to hit the ground?
3-4
© Glencoe/McGraw-Hill 98 Advanced Mathematical Concepts
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
3-4
__ __17 28
__ __ __ __2 29 6 27
__ __ __31 33 14
__ __ __20 11 34
__ __36 7
__ __ __ __ __24 16 1910 4
__18
__ __ __ __35 3 21 8
__ __ __ __13 9 22 5
__ __ __ __ __ __ __ __26 32 30 23 25 12 15 1
1
14
25
2 3
16
4
17
28
5
18
29
6 7
19
26 30
20
8 9
21
33
13 15
27 3231
10
22
34
11
23
35
12
24
36
An Inverse AcrosticThe puzzle on this page is called an acrostic. To solve the puzzle,work back and forth between the clues and the puzzle box. You mayneed a math dictionary to help with some of the clues.
1. If a relation contains the element (e, v), then theinverse of the relation must contain the element( __, __ ).
2. The inverse of the function 2x is found by computing __ of x.
3. The first letter and the last two letters of themeaning of the symbol f –1 are __ .
4. This is the product of a number and its multiplicative inverse.
5. If the second coordinate of the inverse of (x, f (x )) isy, then the first coordinate is read “__ of __”.
6. The inverse ratio of two numbers is the __ of thereciprocals of the numbers.
7. If ? is a binary operation on set S andx ? e 5 e ? x 5 x for all x in S, then an identity element for the operation is __.
8. To solve a matrix equation, multiply each side ofthe matrix equation on the __ by the inversematrix.
9. Two variables are inversely proportional __ theirproduct is constant.
10. The graph of the inverse of a linear function is a __ line.
From President Franklin D. Roosevelt’s inaugural address during the Great Depression; delivered March 4, 1933.
© Glencoe/McGraw-Hill 99 Advanced Mathematical Concepts
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
3-5
Continuity and End Behavior
Functions can be continuous or discontinuous. Graphs thatare discontinuous can exhibit infinite discontinuity, jumpdiscontinuity, or point discontinuity.
Example 1 Determine whether each function is continuousat the given x-value. Justify your answer usingthe continuity test.
a. ƒ(x) 5 2x1 3; x 5 2
(1) The function is defined at x 5 2;ƒ(2) 5 7.
(2) The tables below show that yapproaches 7 as x approaches 2 from the left and that y approaches 7 as x approaches 2 from the right.
(3) Since the y-values approach 7 as x approaches 2 from both sides and ƒ(2) 5 7, the function is continuous at x 5 2.
The end behavior of a function describes what the y2values doasxbecomes greater and greater. In general, the end behavior ofany polynomial function can be modeled by the function made upsolely of the term with the highest power of x and its coefficient.
Example 2 Describe the end behavior of p(x) 5 2x51 2x3
2 4.
Determine ƒ(x) 5 anxn where xn is the term in p(x) with
the highest power of x and an
is its coefficient.
ƒ(x) 5 2x5 xn5 x5 a
n5 21
Thus, by using the table on page 163 of your text, youcan see that when an is negative and n is odd, the endbehavior can be stated as p(x) → 2∞ as x → ∞ and p(x) → ∞ as x → 2∞.
b. ƒ(x) 5 }x2
22
x1
}; x 5 1
Start with the first condition in the continuity test. The function is notdefined at x 5 1 because substituting 1for x results in a denominator of zero. Sothe function is discontinuous at x 51.
c. ƒ(x) 5 5This function fails the second part of thecontinuity test because the values ofƒ(x) approach 1 as x approaches 2 fromthe left, but the values of ƒ(x) approach5 as x approaches 2 from the right.
x . 2
x 5 2
ifif
2x 1 1
x 2 1x y 5 ƒ(x)
2.1 7.2
2.01 7.02
2.001 7.002
x y 5 ƒ(x)
1.9 6.8
1.99 6.98
1.999 6.998
A function is continuous at x 5 c if it satisfies the following three conditions.
(1) the function is defined at c; in other words, ƒ(c) exists;
(2) the function approaches the same y-value to the left and right of x 5 c; and
(3) the y-value that the function approaches from each side is ƒ(c).
© Glencoe/McGraw-Hill 100 Advanced Mathematical Concepts
Continuity and End Behavior
Determine whether each function is continuous at the given x-value. Justify
your answer using the continuity test.
1. y 5 }32x2}; x 5 21 2. y 5 }
x2 1
2x 1 4} ; x 5 1
3. y 5 x3 2 2x 1 2; x 5 1 4. y 5 }xx
2
1
24
} ; x 5 24
Describe the end behavior of each function.
5. y 5 2x5 2 4x 6. y 5 22x6 1 4x4 2 2x 1 1
7. y 5 x4 2 2x3 1 x 8. y 5 24x3 1 5
Given the graph of the function, determine the interval(s) for which the function
is increasing and the interval(s) for which the function is decreasing.
9.
10. Electronics Ohm’s Law gives the relationship between resistance R, voltage
E, and current I in a circuit as R 5 }EI} . If the voltage remains constant but the
current keeps increasing in the circuit, what happens to the resistance?
Practice
NAME _____________________________ DATE _______________ PERIOD ________
3-5
© Glencoe/McGraw-Hill 101 Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Reading MathematicsThe following selection gives a definition of a continuous function as it might be defined in a college-level mathematics textbook. Noticethat the writer begins by explaining the notation to be used for various types of intervals. It is a common practice for college authors to explain their notation, since, although a great deal of the notation is standard, each author usually chooses the notation he or she wishes to use.
Use the selection above to answer these questions.
1. What happens to the four inequalities in the first paragraph when a 5 b?
2. What happens to the four intervals in the first paragraph when a 5 b?
3. What mathematical term makes sense in this sentence?
If f(x) is not ____?__ at x0, it is said to be discontinuous at x0.
4. What notation is used in the selection to express the fact that a number x is contained in the interval I?
5. In the space at the right, sketch the graph of the function f(x) defined as follows:
f(x) 55 if x [ 30, 2
1, if x [ 3 , 146. Is the function given in Exercise 5 continuous on the
interval [0, 1]? If not, where is the function discontinuous?
1}2
1}2
1}2
Enrichment3-5
Throughout this book, the set S, called the domain of definition of a function, will usually be
an interval. An interval is a set of numbers satisfying one of the four inequalities a , x , b,
a # x , b, a , x # b, or a # x # b. In these inequalities, a # b. The usual notations for the
intervals corresponding to the four inequalities are, respectively, (a, b), [a, b), (a, b], and [a, b].
An interval of the form (a, b) is called open, an interval of the form
[a, b) or (a, b] is called half-open or half-closed, and an interval of the form
[a, b] is called closed.
Suppose I is an interval that is either open, closed, or half-open. Suppose ƒ (x) is a function
defined on I and x0 is a point in I. We say that the function f (x) is continuous at the point x0
if
the quantity |ƒ(x) 2 ƒ(x0)| becomes small as x [ I approaches x
0.
© Glencoe/McGraw-Hill 102 Advanced Mathematical Concepts
Critical Points and Extrema
Critical points are points on a graph at which a line drawntangent to the curve is horizontal or vertical. A critical point may bea maximum, a minimum, or a point of inflection. A point ofinflection is a point where the graph changes its curvature. Graphscan have an absolute maximum, an absolute minimum, arelative maximum, or a relative minimum. The general term formaximum or minimum is extremum (plural, extrema).
Example 1 Locate the extrema for the graph of y 5 ƒ(x).Name and classify the extrema of the function.
a.
The function has an absolute maximum at (0, 2). The absolute maximum is the greatest value that a function assumes over its domain.
b.
The function has an absolute minimum at (21, 0). The absolute minimum is the least value that a function assumes over its domain.
By testing points on both sides of a critical point, you candetermine whether the critical point is a relative maximum, arelative minimum, or a point of inflection.
Example 2 The function ƒ(x) 5 2x61 2x4
2 9x2 has a criticalpoint at x 5 0. Determine whether the criticalpoint is the location of a maximum, a minimum,or a point of inflection.
Because 0 is greater than both ƒ(x 2 0.1) and ƒ(x 1 0.1), x 5 0 is the location of a relative maximum.
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
3-6
c.
The relative maximum and minimummay not be the greatest and the least y-value for the domain, respectively, butthey are the greatest and least y-value onsome interval of the domain. The functionhas a relative maximum at (22, 23) and arelative minimum at (0, 25). Because the graph indicates that the functionincreases or decreases without bound as x increases or decreases, there is neitheran absolute maximum nor an absolute minimum.
x x 2 0.1 x 1 0.1 ƒ( x 2 0.1) ƒ( x) ƒ( x 1 0.1)Type of
Critical Point
0 20.1 0.1 20.0899 0 20.0899 maximum
© Glencoe/McGraw-Hill 103 Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Critical Points and Extrema
Locate the extrema for the graph of y 5 ƒ(x). Name and classify the
extrema of the function.
1. 2.
3. 4.
Determine whether the given critical point is the location of a maximum, a
minimum, or a point of inflection.
5. y 5 x2 2 6x 1 1, x 5 3 6. y 5 x2 2 2x 2 6, x 5 1 7. y 5 x4 1 3x2 2 5, x 5 0
8. y 5 x5 2 2x3 2 2x2, x 5 0 9. y 5 x3 1 x2 2 x, x 5 21 10. y 5 2x3 1 4, x 5 0
11. Physics Suppose that during an experiment you launch a toy rocket straight upward from a heightof 6 inches with an initial velocity of 32 feet per second. The height at any time t can be modeled by the function s(t) 5 216t2 1 32t 1 0.5 where s(t) is measured in feet and t is measured in seconds.Graph the function to f ind the maximum height obtained by the rocket before it begins to fall.
3-6
© Glencoe/McGraw-Hill 104 Advanced Mathematical Concepts
"Unreal" EquationsThere are some equations that cannot be graphed on the real-number coordinate system. One example is the equationx2 – 2x 1 2y2 1 8y 1 14 5 0. Completing the squares in x and ygives the equation (x – 1)2 1 2(y 1 2)2 5 –5.
For any real numbers, x and y, the values of (x – 1)2 and 2(y 1 2)2
are nonnegative. So, their sum cannot be –5. Thus, no real values of x and y satisfy the equation; only imaginary values can be solutions.
Determine whether each equation can be graphed on the real-number plane. Write yes or no.
1. (x 1 3)2 1 (y 2 2)2 5 24 2. x2 2 3x 1 y2 1 4y 5 27
3. (x 1 2)2 1 y2 2 6y 1 8 5 0 4. x2 1 16 5 0
5. x4 1 4y2 1 4 5 0 6. x2 1 4y2 1 4xy 1 16 5 0
In Exercises 7 and 8, for what values of k :
a. will the solutions of the equation be imaginary?
b. will the graph be a point?
c. will the graph be a curve?
d. Choose a value of k for which the graph is a curve and sketch the curve on the axes provided.
7. x2 2 4x 1 y2 1 8y 1 k 5 0 8. x2 1 4x 1 y2 2 6y 2 k 5 0
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
3-6
© Glencoe/McGraw-Hill 105 Advanced Mathematical Concepts
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
Graphs of Rational Functions
Example 1 Determine the asymptotes for the graph of ƒ(x) 5 }
2xx
1
2
31
}.
Since ƒ(23) is undefined, there may be a vertical asymptote at x 5 23. To verify that x 5 23 is a vertical asymptote, check to see that ƒ(x) → ∞ or ƒ(x) → 2∞ as x → 23 from either the left or the right.
The values in the table confirm that ƒ(x) → 2∞ as x → 23 from the right, so there is a vertical asymptote at x 5 23.
Example 2 Determine the slant asymptote for
ƒ(x) 5 }3x2
x2
2
2x1
1 2}.
First use division to rewrite the function.
As x → ∞, }x 2
31
} → 0. Therefore, the graph of ƒ(x)
will approach that of y 5 3x 1 1. This means thatthe line y 5 3x 1 1 is a slant asymptote for thegraph of ƒ(x).
3-7
One way to find the horizontalasymptote is to let ƒ(x) 5 y and solve for x in terms of y. Then find where the function isundefined for values of y.
y 5 }2xx1
2
31
}
y(x 1 3) 5 2x 2 1xy 1 3y 5 2x 2 1xy 2 2x 5 23y 2 1x(y 2 2) 5 23y 2 1
x 5 }2
y
3y
2
2
2
1}
The rational expression }2
y
3y
2
2
2
1} is
undefined for y 5 2. Thus, thehorizontal asymptote is the line y 5 2.
x ƒ(x)
22.9 268
22.99 2698
22.999 26998
22.9999 269998
A rational function is a quotient of two polynomial functions.
The line x 5 a is a vertical asymptote for a function ƒ(x) if ƒ(x) → ∞ or ƒ(x) → 2∞ as x → a
from either the left or the right.
The line y 5 b is a horizontal asymptote for a function ƒ(x) if ƒ(x) → b as x → ∞ or as
x → 2∞.
A slant asymptote occurs when the degree of the numerator of a rational function is exactly
one greater than that of the denominator.
3x 1 1x 2 1q3wxw2w2w 2wxw 1w 2w → ƒ(x) 5 3x 1 1 1 }
x 2
31
}
3x2 2 3xx 1 2x 2 1
3
© Glencoe/McGraw-Hill 106 Advanced Mathematical Concepts
Graphs of Rational FunctionsDetermine the equations of the vertical and horizontal asymptotes, if any, of
each function.
1. ƒ(x) 5 }
x2
4
1 1} 2. ƒ(x) 5 }
2
x
x
1
1
1
1} 3. g(x) 5 }
(x 1
x
11
)(x32 2)
}
Use the parent graph ƒ(x) 5 }1x
} to graph each equation. Describe the
transformation(s) that have taken place. Identify the new locations
of the asymptotes.
4. y 5 }x 1
31
} 2 2 5. y 5 2}x 2
43
} 1 3
Determine the slant asymptotes of each equation.
6. y 5 }5x2 2
x 2
102x 1 1} 7. y 5 }
xx
2
1
2
1x
}
8. Graph the function y 5 }x2
x1
1
x12 6
}.
9. Physics The illumination I from a light source is given by the
formula I 5 }d
k2}, where k is a constant and d is distance. As the
distance from the light source doubles, how does the illuminationchange?
Practice
NAME _____________________________ DATE _______________ PERIOD ________
3-7
© Glencoe/McGraw-Hill 107 Advanced Mathematical Concepts
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
Slant AsymptotesThe graph of y 5 ax 1 b, where a Þ 0, is called a slant asymptote of y 5 f (x) if the graph of f (x) comes closer and closer to the line asx Õ ∞ or x Õ – ∞.
For f (x) 5 3x 1 4 1 , y 5 3x 1 4 is a slant asymptote because
f (x) 2 (3x 1 4) 5 , and Õ 0 as x Õ ∞ or x Õ – ∞.
Example Find the slant asymptote of f (x) 5 .
–2 1 8 15 Use synthetic division.–2 –12
1 6 3
y 5 5 x 1 6 1
Since Õ 0 as x Õ ∞ or x Õ –∞,
y 5 x 1 6 is a slant asymptote.
Use synthetic division to find the slant asymptote for each of the following.
1. y 5
2. y 5
3. y 5
4. y 5
5. y 5ax2 1 bx 1 c}}
x 1 d
ax2 1 bx 1 c}}
x 2 d
x2 2 2x 2 18}}
x 2 3
x2 1 3x 2 15}}
x 2 2
8x2 2 4x 1 11}}
x 1 5
3}x 1 2
3}x 1 2
x2 1 8x 1 15}}
x 1 2
x21 8x 1 15
}}x 1 2
2}x
2}x
2}x
3-7
© Glencoe/McGraw-Hill 108 Advanced Mathematical Concepts
Direct, Inverse, and Joint VariationA direct variation can be described by the equation y 5 kxn.The k in this equation is called the constant of variation.To express a direct variation, we say that y varies directly asxn. An inverse variation can be described by the equation
y 5 }xkn} or xny 5 k. When quantities are inversely
proportional, we say they vary inversely with each other.Joint variation occurs when one quantity varies directly asthe product of two or more other quantities and can bedescribed by the equation y 5 kxnzn.
Example 1 Suppose y varies directly as x and y 5 14 when x 5 8.a. Find the constant of variation and write an
equation of the form y 5 kxn.b. Use the equation to find the value of y when
x 5 4.
a. The power of x is 1, so the direct variationequation is y 5 kx.
y 5 kx14 5 k(8) y 5 14, x 5 8
1.75 5 k Divide each side by 8.
The constant of variation is 1.75. Theequation relating x and y is y 5 1.75x.
b. y 5 1.75xy 5 1.75(4) x 5 4y 5 7
When x 5 4, the value of y is 7.
Example 2 If y varies inversely as x and y 5 102 when x 5 7,find x when y 5 12.
Use a proportion that relates the values.
}
x
y1
2
n
} 5 }x
y2
1
n
}
}172} 5 }
10x2
} Substitute the known values.
12x 5 714 Cross multiply.
x 5 }71124
} or 59.5 Divide each side by 12.
When y 5 12, the value of x is 59.5.
Study Guide
NAME _____________________________ DATE _______________ PERIOD ________
3-8
© Glencoe/McGraw-Hill 109 Advanced Mathematical Concepts
Practice
NAME _____________________________ DATE _______________ PERIOD ________
Direct, Inverse, and Joint VariationWrite a statement of variation relating the variables of each equation.
Then name the constant of variation.
1. 2}xy
2} 5 3 2. E 5 IR
3. y 5 2x 4. d 5 6t2
Find the constant of variation for each relation and use it to write an
equation for each statement. Then solve the equation.
5. Suppose y varies directly as x and y 5 35 when x 5 5. Find y when x 5 7.
6. If y varies directly as the cube of x and y 5 3 when x 5 2, find x when y 5 24.
7. If y varies inversely as x and y 5 3 when x 5 25, find x when y 5 10.
8. Suppose y varies jointly as x and z, and y 5 64 when x 5 4 and z 5 8.Find y when x 5 7 and z 5 11.
9. Suppose V varies jointly as h and the square of r, and V 5 45p when r 5 3 and h 5 5. Find r when V 5 175p and h 5 7.
10. If y varies directly as x and inversely as the square of z, and y 5 25 when x 5 10 andz 5 2, find y when x 5 5 and z 5 5.
11. Finances Enrique deposited $200.00 into a savings account. The simple interest I on his account varies jointly as the time t in years and the principal P.After one quarter (three months), the interest on Enrique’s account is $2.75.Write an equation relating interest, principal, and time. Find the constant of variation. Then find the interest after three quarters.
3-8
© Glencoe/McGraw-Hill 110 Advanced Mathematical Concepts
Reading Mathematics: Interpreting Conditional StatementsThe conditional statement below is written in “if-then” form. It has the form p → q where p is the hypothesis and q is the consequent.
If a matrix A has a determinant of 0, then A21 does not exist.
It is important to recognize that a conditional statement need notappear in “if-then” form. For example, the statement
Any point that lies in Quadrant I has a positive x-coordinate.
can be rewritten as
If the point P(x, y) lies in Quadrant I, then x is positive.
Notice that P lying in Quadrant I is a sufficient condition for itsx-coordinate to be positive. Another way to express this is to say that P lying in Quadrant I guarantees that its x-coordinate is positive. On the other hand, we can also say that x being positive is a necessary condition for P to lie in Quadrant I. In other words, Pdoes not lie in Quadrant I if x is not positive.
To change an English statement into “if-then” form requires that you understand the meaning and syntax of the English statement.Study each of the following equivalent ways of expressing p → q.
• If p then q • p implies q• p only if q • only if q, p• p is a sufficient condition for q • not p unless q• q is a necessary condition for p.
Rewrite each of the following statements in “if-then” form.
1. A consistent system of equations has at least one solution.
2. When the region formed by the inequalities in a linear programming application is unbounded, an optimal solution for the problem may not exist.
3. Functions whose graphs are symmetric with respect to the y-axis are called even functions.
4. In order for a decimal number d to be odd, it is sufficient that d end in the digit 7.
Enrichment
NAME _____________________________ DATE _______________ PERIOD ________
3-8
© Glencoe/McGraw-Hill 111 Advanced Mathematical Concepts
Chapter 3 Test, Form 1A
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right of
each problem.
1. The graph of the equation y 5 x3 2 x is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these
2. If the graph of a function is symmetric about the origin, which of the 2. ________following must be true?A. ƒ(x) 5 ƒ(2x) B. ƒ(2x) 5 2ƒ(x) C. ƒ(x) 5 uƒ(x)u D. ƒ(x) 5 }
ƒ(1x)}
3. The graph of an odd function is symmetric with respect to which 3. ________of the following?A. the x-axis B. the y-axis C. the line y 5 x D. none of these
4. Given the parent function p(x) 5 vxb, what transformations 4. ________occur in the graph of p(x) 5 2vx 2 3b 1 4?A. vertical expansion by factor B. vertical compression by factor of
of 2, left 3 units, up 4 units 0.5, down 3 units, left 4 unitsC. vertical expansion by factor D. vertical compression by factor of
of 2, right 3 units, up 4 units 0.5, down 3 units, left 4 units
5. Which of the following results in the graph of ƒ(x) 5 x2 being expanded 5. ________vertically by a factor of 3 and reflected over the x-axis?
A. ƒ(x) 5 }13
}x2 B. ƒ(x) 5 23x2 C. ƒ(x) 5 2}x12} 1 3 D. ƒ(x) 5 2}
13
}x2
6. Which of the following represents the graph of ƒ(x) 5 uxu3 2 1? 6. ________A. B. C. D.
7. Solve u2x 2 5u $ 7. 7. ________A. x # 21 or x $ 6 B. 21 # x # 6C. x # 6 D. x $ 21
8. Choose the inequality shown by the graph. 8. ________A. y # 2uxu 1 1B. y # 22uxu 1 1C. y # 2uxu 2 1D. y # 22uxu 2 1
9. Find the inverse of ƒ(x) 5 2Ïxw 1 3. 9. ________
A. ƒ21(x) 5 1}x 2
23
}22 B. ƒ21(x) 5 1}x 1
23
}22
C. ƒ21(x) 5 }12
}Ïxw 2 3 D. ƒ21(x) 5 }12
}Ïxw 1 3
10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.
Chapter
3
© Glencoe/McGraw-Hill 112 Advanced Mathematical Concepts
11. Describe the end behavior of ƒ(x) 5 2x4 2 5x 1 1. 11. ________A. x → `, ƒ(x)→ ` B. x→ `, ƒ(x) → `
x→ 2`, ƒ(x) → 2` x → 2`, ƒ(x) → `
C. x → `, ƒ(x) → 2` D. x → `, ƒ(x) → 2`
x → 2`, ƒ(x) → ` x → 2`, ƒ(x) → 2`
12. Which type of discontinuity, if any, is shown 12. ________in the graph at the right?A. jump B. infiniteC. point D. The graph is continuous.
13. Choose the statement that is true for the graph of ƒ(x) 5 x3 2 12x. 13. ________A. ƒ(x) increases for x . 22. B. ƒ(x) decreases for x . 22.C. ƒ(x) increases for x . 2. D. ƒ(x) decreases for x , 2.
14. Which type of critical point, if any, is present in the graph of 14. ________ƒ(x) 5 (2x 1 4)5 1 1?A. maximum B. minimumC. point of inflection D. none of these
15. Which is true for the graph of ƒ(x) 5 2x3 1 3x 2 2? 15. ________A. relative maximum at (1, 0) B. relative minimum at (1, 0)C. relative maximum at (21, 24) D. relative minimum at (0, 22)
16. Which is true for the graph of y 5 }xx
2
22
1
49
}? 16. ________
A. vertical asymptotes at x 5 63 B. horizontal asymptotes at y 5 6 2C. vertical asymptotes at x 5 62 D. horizontal asymptote at y 5 1
17. Find the equation of the slant asymptote for the graph of 17. ________
y 5 .
A. y 5 x 2 4 B. y 5 x 1 1 C. y 5 x D. y 5 1
18. Which of the following could be the function 18. ________represented by the graph at the right?
A. ƒ(x) 5 }xx
1
1
12
} B. ƒ(x) 5 }(
(
x
x
1
2
1
3
)
)
(
(
x
x
2
1 2
3
)
)}
C. ƒ(x) 5 }xx
1
2
12
} D. ƒ(x) 5 }(
(
x
x
1
1
3
1
)
)
(
(
x
x
1
1
2
3
)
)}
19. Chemistry The volume V of a gas varies inversely as pressure P is 19. ________exerted. If V 5 3.5 liters when P 5 5 atmospheres, find V when P 5 8 atmospheres.A. 2.188 liters B. 5.600 liters C. 11.429 liters D. 17.5 liters
20. If y varies jointly as x and the cube of z, and y 5 378 when x 5 4 and 20. ________z 5 3, find y when x 5 9 and z 5 2.A. y 5 283.5 B. y 5 84 C. y 5 567 D. y 5 252
Bonus An even function ƒ has a vertical asymptote at x 5 3 and a Bonus: ________maximum at x 5 0. Which of the following could be ƒ ?
A. ƒ(x) 5 }x2 2
x9
} B. ƒ(x) 5 }x 2
x3
} C. ƒ(x) 5 }x2
x1
2
9} D. ƒ(x) 5 }
x4 2
x2
81}
x3 2 4x2 1 2x 2 5}}}
x2 1 2
Chapter 3 Test, Form 1A (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 113 Advanced Mathematical Concepts
Chapter 3 Test, Form 1B
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right of
each problem.
1. The graph of the equation y 5 x4 2 3x2 is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these
2. If the graph of a function is symmetric with respect to the y-axis, 2. ________which of the following must be true?A. ƒ(x) 5 ƒ(2x) B. ƒ(x) 5 2ƒ(x) C. ƒ(x) 5uƒ(x)u D. ƒ(x) 5 }
ƒ(1x)}
3. The graph of an even function is symmetric with respect to which of 3. ________the following?A. the x-axis B. the y-axis C. the line y 5 xD. none of these
4. Given the parent function p(x) 5 x2, what translations occur in 4. ________the graph of p(x) 5 (x 2 7)2 1 3?A. right 7 units, up 3 units B. down 7 units, left 3 unitsC. left 7 units, up 3 units D. right 7 units, down 3 units
5. Which of the following results in the graph of ƒ(x) 5 x3 being expanded 5. ________vertically by a factor of 3?A. ƒ(x) 5 x3 1 3 B. ƒ(x) 5 }
13
}x3 C. ƒ(x) 5 3x3 D. ƒ(x) 5 2}13
}x3
6. Which of the following represents the graph of ƒ(x) 5 ux2 2 1u? 6. ________A. B. C. D.
7. Solve u2x 2 4u , 6. 7. ________A. x , 21 or x . 5 B. 21 , x , 5C. x , 5 D. x . 21
8. Choose the inequality shown by the graph. 8. ________
A. y # ux 1 2u 1 1 B. y # ux 2 2u 1 1
C. y $ ux 1 2u 1 1 D. y $ ux 2 2u 1 1
9. Find the inverse of ƒ(x) 5 }x 2
12
}. 9. ________
A. ƒ21(x) 5 }
x 2
12
} B. ƒ21(x) 5 }
1x
} 1 2
C. ƒ21(x) 5 x 1 2 D. ƒ
21(x) 5 }1x
} 2 2
10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.
Chapter
3
© Glencoe/McGraw-Hill 114 Advanced Mathematical Concepts
11. Which type of discontinuity, if any, is shown 11. ________in the graph at the right?A. jump B. infiniteC. point D. The graph is continuous.
12. Describe the end behavior of ƒ(x) 5 2x3 2 5x 1 1 12. ________A. x → `, ƒ(x) → ` B. x → `, ƒ(x) → `
x → 2`, ƒ(x) → 2` x → 2`, ƒ(x) → `C. x → `, ƒ(x) → 2` D. x → `, ƒ(x) → 2`
x → 2`, ƒ(x) → ` x → 2`, ƒ(x) → 2`
13. Choose the statement that is true for the graph of ƒ(x) 5 2(x 2 2)2. 13. ________A. ƒ(x) increases for x . 22. B. ƒ(x) decreases for x . 22.C. ƒ(x) increases for x , 2. D. ƒ(x) decreases for x , 2.
14. Which type of critical point, if any, is present in the graph of 14. ________ƒ(x) 5 (2x 1 4)3?A. maximum B. minimumC. point of inflection D. none of these
15. Which is true for the graph of ƒ(x) 5 x3 2 3x 1 2? 15. ________A. relative maximum at (1, 0) B. relative minimum at (21, 4)C. relative maximum at (21, 4) D. relative minimum at (0, 2)
16. Which is true for the graph of y 5 }xx
2
22
2
49
}? 16. ________
A. vertical asymptotes, x 5 63 B. horizontal asymptotes, y 5 62C. vertical asymptotes, x 5 62 D. horizontal asymptote, y 5 0
17. Find the equation of the slant asymptote for the graph of 17. ________
y 5 }3x2
x+
2
2x12 3
}.
A. y 5 x B. y 5 3x 1 1 C. y 5 x 1 3 D. y 5 3x 1 5
18. Which of the following could be the function 18. ________represented by the graph at the right?
A. ƒ(x) 5 }x 1
x2
} B. ƒ(x) 5 }(x 2
x(x3)
2
(x31
)2)
}
C. ƒ(x) 5 }x 2
x2
} D. ƒ(x) 5 }(x 1
x(x3)
1
(x31
)2)
}
19. Chemistry The volume V of a gas varies inversely as pressure P is 19. ________exerted. If V 5 4 liters when P 5 3.5 atmospheres, find V when P 5 2.5 atmospheres.A. 5.6 liters B. 2.188 liters C. 2.857 liters D. 10.0 liters
20. If y varies jointly as x and the cube root of z, and y 5 120 when x 5 3 20. ________and z 5 8, find y when x 5 4 and z 5 27.A. y 5 540 B. y 5 240 C. y 5 60 D. y 5 26}
23
}
Bonus If ƒ( g(x)) 5 x and ƒ(x) 5 3x 2 4, find g(x). Bonus: ________
A. g(x) 5 }x 2
34
} B. g(x) 5 }x +
34
} C. g(x) 5 }3x
} 1 4 D. g(x) 5 }3x
} 2 4
Chapter 3 Test, Form 1B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 115 Advanced Mathematical Concepts
Chapter 3 Test, Form 1C
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right of
each problem.
1. The graph of the equation y 5 x2 2 3 is symmetric with respect to 1. ________which of the following?A. the x-axis B. the y-axis C. the origin D. none of these
2. If the graph of a relation is symmetric about the line y 5 x and the 2. ________point (a, b) is on the graph, which of the following must also be on the graph?A. (2a, b) B. (a, 2b) C. (b, a) D. (2a, 2b)
3. The graph of an odd function is symmetric with respect to which 3. ________of the following?A. the x-axis B. the y-axis C. the origin D. none of these
4. Given the parent function p(x) 5 Ïxw, what transformation occurs in 4. ________the graph of p(x) 5 Ïxw 1w 2w 2 5?A. right 2 units, up 5 units B. up 2 units, right 5 units C. left 2 units, down 5 units D. down 2 units, left 5 units
5. Which of the following results in the graph of ƒ(x) 5 x2 being 5. ________expanded vertically by a factor of 4?A. ƒ(x) 5 x2 1 4 B. ƒ(x) 5 x2 2 4 C. ƒ(x) 5 4x2 D. ƒ(x) 5 }
14
}x2
6. Which of the following represents the graph of ƒ(x) 5 ux3u? 6. ________
A. B. C. D.
7. Solve ux 2 4u , 6. 7. ________A. x . 22 B. x , 10C. 22 , x , 10 D. x , 22 or x . 10
8. Choose the inequality shown by the graph. 8. ________A. y # x2 1 1 B. y # (x 2 1)2
C. y # x2 2 1 D. y # (x 1 1)2
9. Find the inverse of ƒ(x) 5 2x 2 4. 9. ________A. ƒ21(x) 5 }
2x12 4} B. ƒ21(x) 5 }
x 1
24
}
C. ƒ21(x) 5 }2x
} 1 4 D. ƒ21(x) 5 }2x
} 2 4
10. Which graph represents a function whose inverse is also a function? 10. ________A. B. C. D.
Chapter
3
© Glencoe/McGraw-Hill 116 Advanced Mathematical Concepts
11. Which type of discontinuity, if any, is shown 11. ________in the graph at the right?A. jumpB. infiniteC. pointD. The graph is continuous.
12. Describe the end behavior of ƒ(x) 5 2x2. 12. ________A. x → ∞, ƒ(x) → ∞ B. x → ∞, ƒ(x) → ∞
x → 2∞, ƒ(x) → 2∞ x → 2∞, ƒ(x) → ∞C. x → ∞, ƒ(x) → 2∞ D. x → ∞, ƒ(x) → 2∞
x → 2∞, ƒ(x) → ∞ x → 2∞, ƒ(x) → 2∞
13. Choose the statement that is true for the graph of ƒ(x) 5 (x 1 1)2. 13. ________A. ƒ(x) increases for x . 21. B. ƒ(x) decreases for x . 21.C. ƒ(x) increases for x , 1. D. ƒ(x) decreases for x , 1.
14. Which type of critical point, if any, is present in the graph 14. ________of ƒ(x) 5 x3 2 1?A. maximum B. minimumC. point of inflection D. none of these
15. Which is true for the graph of ƒ(x) 5 x3 2 3x? 15. ________A. relative maximum at (1, 22) B. relative minimum at (21, 2)C. relative maximum at (21, 2) D. relative minimum at (0, 0)
16. Which is true for the graph of y 5 }xx
2
22
2
94
}? 16. ________
A. vertical asymptotes at x 5 63 B. horizontal asymptotes at y 5 62C. vertical asymptotes at x 5 62 D. horizontal asymptote at y 5 0
17. Find the equation of the slant asymptote for the graph of y 5 }xx
2
1
2
35
}. 17. ________
A. x 5 3 B. y 5 x 2 3 C. y 5 x D. y 5 x 1 3
18. Which of the following could be the function 18. ________represented by the graph?A. ƒ(x) 5 }
x 2
14
} B. ƒ(x) 5 }(x 2
x22
)(x21 4)
}
C. ƒ(x) 5 }xx
1
2
24
} D. ƒ(x) 5 }(x 2
x22
)(x22 4)
}
19. Chemistry The volume V of a gas varies inversely as pressure 19. ________P is exerted. If V 5 4 liters when P 5 3 atmospheres, find V when P 5 7 atmospheres.A. 1.714 liters B. 5.25 liters C. 9.333 liters D. 1.5 liters
20. If y varies inversely as the cube root of x, and y 5 12 when x 5 8, 20. ________find y when x 5 1.A. y 5 6144 B. y 5 24 C. y 5 6 D. y 5 }
1238
}
Bonus The graph of ƒ(x) 5 }x2
12 c} has a vertical asymptote at x 5 3. Bonus: ________
Find c.A. 9 B. 3 C. 23 D. 29
Chapter 3 Test, Form 1C (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 117 Advanced Mathematical Concepts
Chapter 3 Test, Form 2A
NAME _____________________________ DATE _______________ PERIOD ________
Determine whether the graph of each equation is symmetric with
respect to the origin, the x-axis, the y-axis, the line y 5 x, the line
y 5 2x, or none of these.
1. xy 5 24 1. __________________
2. x 5 5y2 2 2 2. __________________
3. Determine whether the function ƒ(x) 5 }x2 2
x4
} is odd, even, 3. __________________or neither.
4. Describe the transformations relating the graph of 4. __________________y 5 22x3 1 4 to its parent function, y 5 x3.
5. Use transformations of the parent graph p(x) 5 }1x
} 5.
to sketch the graph of p(x) 5 }u1xu} 2 1.
6. Graph the inequality y . 2x2 2 1. 6.
7. Solve u5 2 2xu $ 11. 7. __________________
Find the inverse of each function and state whether the
inverse is a function.
8. ƒ(x) 5 }x 1
x2
} 8. __________________
9. ƒ(x) 5 x2 2 4 9. __________________
10. Graph ƒ(x) 5 x3 2 2 and its inverse. State whether the 10. __________________inverse is a function.
Chapter
3
© Glencoe/McGraw-Hill 118 Advanced Mathematical Concepts
Determine whether each function is continuous at the given
x-value. If discontinuous, state the type of discontinuity (point,
jump, or infinite).
11. ƒ(x) 5 5 ; x 5 1 11. __________________
12. ƒ(x) 5 }xx
2
1
1
39
}; x 5 23 12. __________________
13. Describe the end behavior of y 5 23x4 2 2x. 13. __________________
14. Locate and classify the extrema for the graph of 14. __________________y 5 x4 2 3x2 1 2.
15. The function ƒ(x) 5 x3 2 3x2 1 3x has a critical point 15. __________________when x 5 1. Identify the point as a maximum, a minimum, or a point of inf lection, and state its coordinates.
16. Determine the vertical and horizontal asymptotes for 16. __________________
the graph of y 5 }x3 2
x2
52
x241 6x
}.
17. Find the slant asymptote for y 5 }3x2
x2
2
5x2
1 1}. 17. __________________
18. Sketch the graph of y 5 }x3 2
x2
x22
2
112x
}. 18.
19. If y varies directly as x and inversely as the square root 19. __________________of z, and y 5 8 when x 5 4 and z 5 16, find y when x 5 10 and z 5 25.
20. Physics The kinetic energy Ek of a moving object, 20. __________________
measured in joules, varies jointly as the mass mof the object and the square of the speed v. Find the constant of variation k if E
kis 36 joules, m is 4.5 kilograms,
and v is 4 meters per second.
Bonus Given the graph of p(x), Bonus:sketch the graph of
y 5 22p3}12
}(x 2 2)4 1 2.
x2 1 1 if x , 12x3 1 2 if x $ 1
Chapter 3 Test, Form 2A (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 119 Advanced Mathematical Concepts
Chapter 3 Test, Form 2B
NAME _____________________________ DATE _______________ PERIOD ________
Determine whether the graph of each equation is symmetric with
respect to the origin, the x-axis, the y-axis, the line y 5 x, the line
y 5 2x, or none of these.
1. xy 5 2 1. __________________
2. y 5 5x3 2 2x 2. __________________
3. Determine whether the function ƒ(x) 5 uxu is odd, even, 3. __________________or neither.
4. Describe the transformations relating the graph of 4. __________________y 5 }
12
}(x 2 3)2 to its parent function, y 5 x2.
5. Use transformations of the parent graph of p(x) 5 Ïxw 5.to sketch the graph of p(x) 5 2Ïxw 2 3.
6. Graph the inequality y # (x 2 2)3. 6.
7. Solve u2x 2 4u # 10. 7. __________________
Find the inverse of each function and state whether the
inverse is a function.
8. ƒ(x) 5 x3 2 4 8. __________________
9. ƒ(x) 5 (x 2 1)2 9. __________________
10. Graph ƒ(x) 5 x2 1 2 and its inverse. State whether the 10. __________________inverse is a function.
Chapter
3
© Glencoe/McGraw-Hill 120 Advanced Mathematical Concepts
Determine whether each function is continuous at the given
x-value. If discontinuous, state the type of discontinuity (point,
jump, or infinite).
11. ƒ(x) 5 5 ; x 5 1 11. __________________
12. ƒ(x) 5 }xx
2
1
2
39
}; x 5 23 12. __________________
13. Describe the end behavior of y 5 3x3 2 2x. 13. __________________
14. Locate and classify the extrema for the graph 14. __________________of y 5 x4 2 3x2.
15. The function ƒ(x) 5 2x3 2 6x2 2 12x 2 7 has a critical 15. __________________point when x 5 22. Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.
16. Determine the vertical and horizontal asymptotes for the 16. __________________
graph of y 5 }xx3
2
2
2
x42}.
17. Find the slant asymptote for y 5 }2x2
x2
2
5x3
1 2}. 17. __________________
18. Sketch the graph of y 5 }xx2 2
2
31x
}. 18.
19. If y varies directly as x and inversely as the square of z, 19. __________________and y 5 8 when x 5 4 and z 5 3, find y when x 5 6 and z 5 22.
20. Geometry The volume V of a sphere varies directly as 20. __________________the cube of the radius r. Find the constant of variation kif V is 288p cubic centimeters and r is 6 centimeters.
Bonus Determine the value of k such that Bonus: __________________
ƒ(x) 5 5 is continuous when x 5 2.2x2 if x , 2x 1 k if x $ 2
x , 1 x $ 1
ifif
x2 1 1 2x 1 3
Chapter 3 Test, Form 2B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 121 Advanced Mathematical Concepts
Chapter 3 Test, Form 2C
NAME _____________________________ DATE _______________ PERIOD ________
Determine whether the graph of each equation is symmetric with
respect to the origin, the x-axis, the y-axis, the line y 5 x, the line
y 5 2x, or none of these.
1. y 5 2uxu 1. __________________
2. y 5 x3 2 x 2. __________________
3. Determine whether the function ƒ(x) 5 x2 1 2 is odd, even, 3. __________________or neither.
4. Describe the transformation relating the graph of 4. __________________y 5 (x 2 1)2 to its parent function, y 5 x2.
5. Use transformations of the parent graph p(x) 5 x3 5.to sketch the graph of p(x) 5 (x 1 2)3 2 1.
6. Graph the inequality y $ x2 2 1. 6.
7. Solve ux 2 4u # 8. 7. __________________
Find the inverse of each function and state whether the
inverse is a function.
8. ƒ(x) 5 x2 8. __________________
9. ƒ(x) 5 x3 2 1 9. __________________
10. Graph ƒ(x) 5 22x 1 4 and its inverse. State whether the 10. __________________inverse is a function.
Chapter
3
© Glencoe/McGraw-Hill 122 Advanced Mathematical Concepts
Determine whether each function is continuous at the given
x-value. If discontinuous, state the type of discontinuity (point,
jump, or infinite).
11. ƒ(x) 5 5 ; x 5 0 11. __________________
12. ƒ(x) 5 }x
x2
1
1
39
} ; x 5 23 12. __________________
13. Describe the end behavior of y 5 x4 2 x2. 13. __________________
14. Locate and classify the extrema for the graph of 14. __________________y 5 2x4 2 2x2.
15. The function ƒ(x) 5 x3 2 3x has a critical point when x 5 0. 15. __________________Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.
16. Determine the vertical and horizontal asymptotes for the 16. __________________
graph of y 5 }x
x2 2
2
245
}.
17. Find the slant asymptote for y 5 }x2
x2
2
x11 2
}. 17. __________________
18. Sketch the graph of y 5 }x2
x2 4}. 18.
19. If y varies directly as the square of x, and y 5 200 19. __________________when x 5 5, find y when x 5 2.
20. If y varies inversely as the cube root of x, and y 5 10 20. __________________when x 5 27, find y when x 5 8.
Bonus Determine the value of k such that Bonus: __________________ƒ(x) 5 3x2 1 kx 2 4 is an even function.
x2 1 1 if x , 02 x if x $ 0
Chapter 3 Test, Form 2C (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
© Glencoe/McGraw-Hill 123 Advanced Mathematical Concepts
Chapter 3 Open-Ended Assessment
NAME _____________________________ DATE _______________ PERIOD ________
Instructions: Demonstrate your knowledge by giving a clear, concise
solution to each problem. Be sure to include all relevant drawings and
justify your answers. You may show your solution in more than one way
or investigate beyond the requirements of the problem.
1. a. Draw the parent graph and each of the three transformations that would result in the graph shown below. Write the equation for each graph and describe the transformation.
b. Draw the parent graph of a polynomial function. Then draw a transformation of the graph. Describe the type of transformation you performed. Use equations as needed to clarify your answer.
c. Sketch the graph of y 5 }x(x 1
x32
)(x4
2 4)}. Describe how you
determined the shape of the graph and its vertical asymptotes.
d. Write the equation of a graph that has a vertical asymptote at x 5 2 and point discontinuity at x 5 21. Describe your reasoning.
2. The critical points of ƒ(x) 5 }15
}x5 2 }14
}x4 2 }23
}x3 1 2 are at x 5 0,
2, and 21. Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection.Explain why x 5 3 is not a critical point of the function.
3. Mrs. Custer has 100 bushels of soybeans to sell. The current price for soybeans is $6.00 a bushel. She expects the market price of a bushel to rise in the coming weeks at a rate of $0.10 per week. For each week she waits to sell, she loses 1 bushel due to spoilage.
a. Find a function to express Mrs. Custer’s total income from selling the soybeans. Graph the function and determine when Mrs. Custer should sell the soybeans in order to maximizeher income. Justify your answer by showing that selling ashort time earlier or a short time later would result in lessincome.
b. Suppose Mrs. Custer loses 5 bushels per week due to spoilage. How would this affect her selling decision?
Chapter
3
© Glencoe/McGraw-Hill 124 Advanced Mathematical Concepts
Determine whether the graph of each equation is
symmetric with respect to the origin, the x-axis, the
y-axis, the line y 5 x, the line y 5 2x, or none of these.
1. x 5 y2 2 1 1. __________________
2. xy 5 8 2. __________________
3. Determine whether the function ƒ(x) 5 2(x 2 1)2 1 4x is 3. __________________odd, even, or neither.
4. Describe the transformations that relate the graph 4. __________________of p(x) 5 (0.5x)3 1 1 to its parent function p(x) 5 x3.
5. Use transformations of the parent graph p(x) 5 vxb 5.to sketch the graph of p(x) 5 2vx 2 3b.
6. Graph the inequality y # Ïxw 2w 1w. 6.
7. Solve ux 2 4u $ 6. 7. __________________
Find the inverse of each function and state whether the
inverse is a function.
8. ƒ(x) 5 x3 1 1 8. __________________
9. ƒ(x) 5 2}1x
} 9. __________________
10. Graph ƒ(x) 5 (x 2 3)2 and its inverse. State whether 10. __________________the inverse is a function.
Chapter 3 Mid-Chapter Test (Lessons 3-1 through 3-4)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
3
Determine whether the graph of each equation is
symmetric with respect to the origin, the x-axis, the
y-axis, the line y 5 x, the line y 5 2x, or none of these.
1. y 5 (x 2 2)3 2. y 5 }x
14} 1. __________________
2. __________________
3. Determine whether the function ƒ(x) 5 x3 2 4x is 3. __________________odd, even, or neither.
4. Describe how the graphs of ƒ(x) 5 uxu and g(x) 5 24uxu 4. __________________are related.
5. Use transformations of the parent graph of p(x) 5 x2 to 5.sketch the graph of p(x) 5 (x 2 1)2 2 2.
1. Graph y . 2x2 1 1. 1.
2. Solve ux 1 3u . 4. 2. __________________
3. Find the inverse of ƒ(x) 5 3x 2 6. Is the inverse a function? 3. __________________
4. Graph ƒ(x) 5 (x 2 1)2 1 2 and its inverse. 4.
5. What is the equation of the line that acts as a line of 5. __________________symmetry between the graphs of ƒ(x) and ƒ21(x)?
Chapter 3 Quiz B (Lessons 3-3 and 3-4)
NAME _____________________________ DATE _______________ PERIOD ________
Chapter 3 Quiz A (Lessons 3-1 and 3-2)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 125 Advanced Mathematical Concepts
Chapter
3
Chapter
3
Determine whether the function is continuous at the given x-value. If
discontinuous, state the type of discontinuity ( jump, infinite, or point).
1. ƒ(x) 5 5 ; x 5 0 1. __________________
2. ƒ(x) 5 }x2
12 1} ; x 5 1 2. __________________
3. Describe the end behavior of y 5 x4 2 3x 1 1. 3. __________________
4. Locate and classify the extrema for the graph of 4. __________________y 5 2x3 2 6x.
5. The function ƒ(x) 5 }x5
5} 2 }
43
}x3 has a critical point at x 5 2. 5. __________________
Identify the point as a maximum, a minimum, or a point of inflection, and state its coordinates.
1. Determine the equations of the vertical and horizontal 1. __________________
asymptotes, if any, for the graph of y 5 }x2 2
x5
2
x 1 6} .
2. Find the slant asymptote for ƒ(x) 5 }x2 2
x 2
4x21 3
}. 2. __________________
3. Use the parent graph ƒ(x) 5 }1x
} to graph the function 3. __________________
g(x) 5 }x 2
23
}. Describe the transformation(s) that have
taken place. Identify the new locations of the asymptotes.
Find the constant of variation for each relation and use it to
write an equation for each statement. Then solve the equation.
4. If y varies directly as x, and y 5 12 when x 5 8, find y when 4. __________________x 5 14.
5. If y varies jointly as x and the cube of z, and y 5 48 when 5. __________________x 5 3 and z 5 2, find y when x 5 22 and z 5 5.
x2 + 1 if x , 0x if x $ 0
Chapter 3 Quiz D (Lessons 3-7 and 3-8)
NAME _____________________________ DATE _______________ PERIOD ________
Chapter 3 Quiz C (Lessons 3-5 and 3-6)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 126 Advanced Mathematical Concepts
Chapter
3
Chapter
3
© Glencoe/McGraw-Hill 127 Advanced Mathematical Concepts
Chapter 3 SAT and ACT Practice
NAME _____________________________ DATE _______________ PERIOD ________
After working each problem, record the
correct answer on the answer sheet
provided or use your own paper.
Multiple Choice
1. If x 5 y 2 4, what is the value of uy 2 xu 1 ux 2 yu?A 0B 4C 8D 16E It cannot be determined from the
information given.
2. If 2x 5 5 2 4y and 12y 5 6x 2 15, thenwhich of the following is true?A 2y 5 5B 2x 5 5C y 1 2x , 5D x 1 2y . 5E 2y 2 x . 5
3. If x 5 }45
}, then 3(x 2 1) 1 4x is how
many fifths?A 13B 15C 17D 23E 25
4. If x2 1 3y 5 10 and }xy
} 5 21, then
which of the following could be a value of x?
A 25B 22C 21D 2E 3
5. 168:252 as 6: ?A 4B 5C 9D 18E 36
6. (3.1 2 12.4) 2 (3.1 1 12.4) 5A 224.8B 26.2C 0D 6.2E 24.8
7. If x 5 5z and y 5 }15z
1+ 2}, then which of
the following is equivalent to y?
A }x +
52
}
B }31x}
C }x 1
12
}
D }3x
11 2}
E None of these
8. If }x 1
x
y} 5 3 and }
z 1
z
y} 5 5, then which
of the following is the value of }xz
}?
A }12
}
B }53
}
C }53
}
D 2E 15
9. !1§ 1§ }12§}§1§ }
14§}§1§ }
18§}§1§ }
1§16}§ 1§ }
3§12}§ 5
A }3Ï
86w
}
B }Ï
86w2w}
C }Ï42w}
D }14
}
E }3Ï
81w4w}
Chapter
3
The larger valueof x in theequation
x2 1 7x 1 12 5 0
The smallervalue of x in the
equationx2 2 7x 1 12 5 0
© Glencoe/McGraw-Hill 128 Advanced Mathematical Concepts
Chapter 3 SAT and ACT Practice (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
310. If 12 and 18 each divide R without a
remainder, which of the following could be the value of R?A 24B 36C 48D 54E 120
11. If x 1 y 5 5 and x 2 y 5 3, then x2 2 y2 5
A 2B 4C 8D 15E 16
12. If n , 0, then which of the followingstatements are true?
I. Ï3
2wnw , 0
II. Ï1w 2w nw , 1
III. Ï5
nw3w . 0A I onlyB III onlyC I and III onlyD II and III onlyE None of the statements are true.
13. If x* is defined to be x 2 2, which of thefollowing is the value of (3* 1 4*)*?A 0B 1C 2D 3E 4
14. Which of the following is the total cost
of 3}12
} pounds of apples at $0.56 per
pound and 4}12
} pounds of bananas at
$0.44 per pound?A $2.00B $3.00C $3.94D $4.00E $4.06
15. If a and b are real numbers where a , b and a 1 b , 0, which of the following MUST be true?A a , 0B b , 0C uau , ubuD ab . 0E It cannot be determined from the
information given.
16. For all x, (3x3 2 13x2 1 6x 2 8)(x 2 4) 5
A 3x4 2 25x3 2 46x2 1 16x 2 32B 3x4 2 25x3 2 46x2 2 32x 1 32C 3x4 2 x3 2 46x2 1 16x 2 32D 3x4 2 25x3 1 58x2 2 32x 1 32E 3x4 2 25x3 1 58x2 1 16x 1 32
17–18. Quantitative ComparisonA if the quantity in Column A is
greaterB if the quantity in Column B is
greaterC if the two quantities are equalD if the relationship cannot be
determined from the informationgiven
Column A Column B
17.
18. (2x 2 3)2 4x(x 2 3)
19. Grid-In For nonzero real numbers a, b, c, and d, d 5 2a, a 5 2b, and
b 5 2c. What is the value of }dc
}?
20. Grid-In If 5 }34
}, what is the
value of x?
3}}
4 1 }x 1
x1
}
© Glencoe/McGraw-Hill 129 Advanced Mathematical Concepts
Chapter 3, Cumulative Review (Chapters 1-3)
NAME _____________________________ DATE _______________ PERIOD ________
1. State the domain and range of {(22, 2), (0, 2), (2, 2)}. Then 1. __________________state whether the relation is a function. Write yes or no.
2. If ƒ(x) 5 }x 2
31
} and g(x) 5 x 1 1, find ƒ( g (x)). 2. __________________
3. Write the standard form of the equation of the line that is 3. __________________parallel to the line with equation y 5 2x 2 3 and passes through the point at (21, 5).
4. Graph ƒ(x) 5 vx 1 1b. 4.
Solve each system of equations algebraically.
5. 3x 1 5y 5 21 5. __________________x 1 y 5 5
6. x 2 2y 1 z 5 7 6. __________________3x 1 y 2 z 5 22x 1 3y 1 2z 5 7
7. How many solutions does a consistent and independent 7. __________________system of linear equations have?
8. Find the value of * *. 8. __________________
9. Determine whether the graph of y 5 x3 2 2x is symmetric 9. __________________with respect to the origin, the x-axis, the y-axis, or none of these.
10. Describe the transformations relating the graph of 10. __________________y 5 2(x 2 1)2 to the graph of y 5 x2.
11. Solve u3x 2 6u , 9. 11. __________________
12. Find the inverse of the function ƒ(x) 5 }x 2
32
}. 12. __________________
13. State the type of discontinuity ( jump, infinite, or point) 13. __________________that is present in the graph of ƒ(x) 5 vxb.
14. Determine the horizontal asymptote for the graph of 14. __________________
ƒ(x) 5 }xx
2
1
35
}.
15. If y varies inversely as the square root of x, and y 5 20 15. __________________when x 5 9, find y when x 5 16.
2722
35
Chapter
3
© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts
SAT and ACT Practice Answer Sheet(10 Questions)
NAME _____________________________ DATE _______________ PERIOD ________
0 0 0
.. .
/ /
.
99 9 9
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
© Glencoe/McGraw-Hill A2 Advanced Mathematical Concepts
SAT and ACT Practice Answer Sheet(20 Questions)
NAME _____________________________ DATE _______________ PERIOD ________
0 0 0
.. .
/ /
.
99 9 9
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
0 0 0
.. .
/ /
.
99 9 9
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
© Glencoe/McGraw-Hill
88
Advanced Mathematical Concepts
Symmetry and Coordinate Graphs
Determine whether the graph of each function is symmetric with respect
to the origin.
1.ƒ(x
) !"
x12!
yes
2.ƒ(x
) x
5"
2no
3.ƒ(x
) x
3"
4x
yes
4.ƒ(x
) ! 3x "
2
x!
no
Determine whether the graph of each equation is symmetric with respect to
the x-axis, the y-axis, the line y�x, the line y� x, or none of these.
5.x#y
6y x
6.x
2#y
2y-axis
7.xy
3y x andy !x
8.x
3#y
2
4x-axis
9.y
4x
none of these
10.y x
2"
1y-axis
11.Is
ƒ(x
)
x a
n e
ven
funct
ion, an o
dd funct
ion, or
nei
ther
?even
Refer to the graph at the right for Exercises 12 and 13.
12.C
omple
te t
he
gra
ph s
o th
at
it is
the
gra
ph
of a
n o
dd funct
ion.
13.C
omple
te t
he
gra
ph s
o th
at
it is
the
gra
ph
of a
n e
ven
funct
ion.
14.Geometry
Cam
eron
tol
d h
er fri
end J
uanit
a
that
the
gra
ph o
f y
6"
3x
has
the
shape
of a
geo
met
ric
figure
. D
eter
min
e w
het
her
the
gra
ph o
f y
6"
3x
is
sym
met
ric
wit
h
resp
ect
to t
he x-a
xis
, th
e y-a
xis
, bot
h, or
nei
ther
. T
hen
make
a s
ket
ch o
f th
e gra
ph.
Is C
am
eron
cor
rect
? x-axis and y-axis;
yes, the graph is that of a diamond or
parallelogram.
Practice
NAME_____________________________ DATE_______________ PERIOD________
3-1
13
12
x
fx() O
(3,3)
(4,2)
(1,1)
Answers (Lesson 3-1)
© Glencoe/McGraw-Hill A3 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
89
Ad
van
ced
Math
em
atical C
on
cep
ts
En
ric
hm
en
t
NAME_____________________________ DATE_______________ PERIOD________
3-1
Symmetry in Three-Dimensional Figures
Aso
lid fig
ure
that
can b
e su
per
impos
ed, poi
nt
for
poi
nt,
on its
mir
ror
image
has
a plane of symmetry
.A
sym
met
rica
l so
lid o
bje
ct m
ay h
ave
a fin
ite
orin
finit
e num
ber
of pla
nes
of sy
mm
etry
. T
he
chair
in t
he
illu
stra
tion
at
the
right
has
just
one
pla
ne
ofsy
mm
etry
; th
e dou
ghnut
has
infinit
ely m
any
pla
nes
of sy
mm
etry
, th
ree
of w
hic
h a
re s
how
n.
Determine the number of planes of symmetry for each object and
describe the planes.
1.
a b
rick
3 p
lan
es o
f sym
me
try;
ea
ch
pla
ne
is p
ara
lle
l to
a p
air
of
op
po
sit
e f
ac
es.
2.
a t
ennis
ball
An
in
fin
ite
nu
mb
er
of
pla
ne
s; e
ac
h p
lan
e p
asse
sth
rou
gh
th
e c
en
ter.
3.
a s
oup c
an
an
in
fin
ite
nu
mb
er
of
pla
ne
s p
assin
g t
hro
ug
hth
e c
en
tra
l a
xis
, p
lus o
ne
pla
ne
cu
ttin
g t
he
ce
n-
ter
of
the
axis
at
rig
ht
an
gle
s4.
a s
quare
pyra
mid
4 p
lan
es a
ll p
assin
g t
hro
ug
h t
he
to
p v
ert
ex:
2 p
lan
es a
re p
ara
lle
l to
a p
air
of
op
po
sit
e e
dg
es
of
the
ba
se
an
d t
he
oth
er
2 c
ut
alo
ng
th
e d
iag
o-
na
ls o
f th
e s
qu
are
ba
se
.5.
a c
ube
9 p
lan
es: 3 p
lan
es a
re p
ara
lle
l to
pa
irs o
f o
pp
o-
sit
e f
ac
es a
nd
th
e o
the
r 6 p
ass t
hro
ug
h p
air
s o
fo
pp
osit
e e
dg
es.
Sol
id fig
ure
s ca
n a
lso
have rotational symmetry
. F
or e
xam
ple
, th
eaxis
dra
wn t
hro
ugh t
he
cube
in t
he
illu
stra
tion
is
a fou
rfol
d a
xis
of
sym
met
ry b
ecause
the
cube
can b
e ro
tate
d a
bou
t th
is a
xis
into
fou
rdiffe
rent
pos
itio
ns
that
are
exact
ly a
like.
6.
How
many fou
r-fo
ld a
xes
of sy
mm
etry
doe
s a c
ube
have?
Use
a d
ie t
o hel
p y
ou loc
ate
them
.
3; e
ac
h a
xis
pa
sse
s t
hro
ug
h t
he
ce
nte
rs o
f a
pa
ir o
f o
pp
osit
e f
ac
es.
7.
Acu
be
has
6 t
wo-
fold
axes
of sy
mm
etry
. In
the
space
at
the
right,
dra
w o
ne
of t
hes
e axes
.
Answers (Lesson 3-2)
© Glencoe/McGraw-Hill A4 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
91
Ad
van
ced
Math
em
atical C
on
cep
ts
Pra
cti
ce
NAME_____________________________ DATE_______________ PERIOD________
Families of Graphs
Describe how the graphs of ƒ(x) and g(x) are related.
1.ƒ(x
) x
2and g
(x)
(x#
3)2"
12.ƒ(x
) x
and g
(x) "2x
g(x
) is
th
e g
rap
h o
f ƒ
(x)
g(x
) is
th
e g
rap
h o
f ƒ
(x)
refl
ec
ted
tra
nsla
ted
le
ft 3
un
its a
nd
o
ve
r th
e x
-axis
an
d c
om
pre
sse
dd
ow
n 1
un
it.
ho
rizo
nta
lly b
y a
fa
cto
r o
f 0
.5.
Use the graph of the given parent function to describe the graph of each
related function.
3.ƒ(x
) x
34.ƒ(x
)
"x#
a.y
2x
3a.y "
x###
3##
1
exp
an
de
d v
ert
ica
lly b
y
tra
nsla
ted
le
ft 3
un
its a
nd
up
a
fa
cto
r o
f 2
1 u
nit
b.y "
0.5
(x"
2)3
b.y
""#x#"
2
refl
ec
ted
ove
r th
e x
-axis
, re
fle
cte
d o
ve
r th
e y
-axis
, tr
an
sla
ted
rig
ht
2 u
nit
s,
tra
nsla
ted
do
wn
2 u
nit
sc
om
pre
sse
d v
ert
ica
lly b
y
a f
ac
tor
of
0.5
c.y (x#
1)3
c.y
"0#.
2#5#x#"
4
tra
nsla
ted
le
ft 1
un
it,
exp
an
de
d h
ori
zo
nta
lly
po
rtio
n b
elo
w t
he
x-a
xis
b
y a
fa
cto
r o
f 4
, re
fle
cte
d s
o t
ha
t it
is
tra
nsla
ted
do
wn
4 u
nit
sa
bo
ve
th
e x
-axis
Sketch the graph of each function.
5.ƒ(x
) "
(x"
1)2# 1
6.ƒ(x
)
2x#
2"
3
7.Consumer Costs
Duri
ng h
er fre
e ti
me,
Jill baby-s
its
the
nei
ghbor
hoo
d c
hildre
n. She
charg
es $
4.5
0 for
each
w
hol
e hou
r or
any fra
ctio
n o
f an h
our. W
rite
and g
raph
a funct
ion t
hat
show
s th
e co
st o
f x
hou
rs o
f baby-s
itti
ng.
f(x)
$4.5
if �x x
4.5
�x "
1
if �x #x
3-2
© G
len
co
e/M
cG
raw
-Hill
92
Ad
van
ced
Math
em
atical C
on
cep
ts
En
ric
hm
en
t
NAME_____________________________ DATE_______________ PERIOD________
3-2
Agraph G′is isomorphic to a graph Gif the following conditions hold.
1.
Gand G′have the same number of vertices and edges.
2.The degree of each vertex in Gis the same as the degree of each corresponding vertex in G′.
3.If two vertices in Gare joined by k(k%0) edges, then the two corresponding vertices in G′are
also joined by kedges.
ba
c
Isomorphic Graphs
AgraphG
is a
col
lect
ion o
f poi
nts
in w
hic
h a
pair
of poi
nts
, ca
lled
vertices, are
co
nnec
ted b
y a
set
of se
gm
ents
or
arc
s, c
alled
edges. T
he degree
of v
erte
x C
, den
oted
deg
(C
), is
the
num
ber
of ed
ges
con
nec
ted t
o th
at
ver
tex. W
e sa
y t
wo
gra
phs
are
isomorphic if th
ey h
ave
the
sam
e st
ruct
ure
. T
he
def
init
ion b
elow
will hel
p y
oudet
erm
ine
whet
her
tw
o gra
phs
are
iso
mor
phic
.
Example
In the graphs below HIJKLMN↔TUVWXYZ.
Determine w
hether the graphs are isomorphic.
Num
ber
of ver
tice
s in
G: 7
Num
ber
of ver
tice
s in
G′:
7N
um
ber
of ed
ges
in G
: 10
Num
ber
of ed
ges
in G′:
10
deg
(H
): 3
deg
(I)
: 3
deg
(T
): 1
deg
(U): 3
deg
(J): 3
deg
(K
): 3
deg
(V
): 3
deg
(W
): 3
deg
(L
): 4
deg
(M
): 3
deg
(X
): 4
deg
(Y
): 3
deg
(N
): 1
deg
(Z
): 3
Sin
ce ther
eare
the
sam
e num
ber
of ve
rtic
es a
nd
the
sam
enum
ber
of e
dge
sand t
her
e are
fiv
e ver
tice
s of
deg
ree
3, on
e ver
tex o
f deg
ree
4, and o
ne
ver
tex o
f deg
ree
1 in b
oth g
raphs,
we
can a
ssum
e th
ey a
re iso
mor
phic
.
Each graph in Row A is isomorphic to one graph in Row B.
Match the graphs that are isomorphic.
Row A
1.
2.
3.
Row B
a.
b.
c.
Answers (Lesson 3-3)
© Glencoe/McGraw-Hill A5 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
94
Ad
van
ced
Math
em
atical C
on
cep
ts
Graphs of Nonlinear Inequalities
Determine whether the ordered pair is a solution for the given inequality.
Write yes or no.
1.y&
(x#
2)2#
3, ("
2, 6)2.y'
(x"
3)3#
2, (4
, 5)
3.y(
2x"
4 "
1, ("
4, 1)
ye
sn
oye
s
Graph each inequality.
4.y(
2 x"
1
5.y&
2(x"
1)2
6.y'
"x#"#
2##
17.y%
(x#
3)3
Solve each inequality.
8.
4x"
10 (
69.
x#
5 #
2 &
610.
2x"
2 "
1 '
7
{x!1$x$
4}
{x!x#!
9 o
r x%!
1}
{x!!
3 #x#
5}
11.Measurement
Inst
ruct
ions
for
buildin
g a
bir
dhou
se w
arn
that
the
pla
tfor
m, w
hic
h idea
lly m
easu
res
14.7
5 c
m2, sh
ould
not
vary
in s
ize
by m
ore
than 0
.30 c
m2. If
it
doe
s, t
he
pre
const
ruct
ed r
oof fo
r th
e bir
dhou
se w
ill not
fit
pro
per
ly.
a.
Wri
te a
n a
bso
lute
valu
e in
equality
that
repre
sents
the
range
of
pos
sible
siz
es for
the
pla
tfor
m. T
hen
sol
ve
for x
to f
ind t
he
range.
!x!
14
.75
!$
0.3
0; {x
!14
.45
$x$
15
.05
}
b.
Den
a c
ut
a b
oard
14.4
2 c
m2. D
oes
the
pla
tfor
m t
hat
Den
a c
ut
fit
wit
hin
the
acc
epta
ble
range?
no
Pra
cti
ce
NAME_____________________________ DATE_______________ PERIOD________
3-3
© G
len
co
e/M
cG
raw
-Hill
95
Ad
van
ced
Math
em
atical C
on
cep
ts
En
ric
hm
en
t
NAME_____________________________ DATE_______________ PERIOD________
Some Parametric Graphs
For
som
e cu
rves
, th
e co
ordin
ate
s x
andy
can b
e w
ritt
en a
s fu
nct
ions
of a
thir
d v
ari
able
. T
he
condit
ions
det
erm
inin
g t
he
curv
e are
giv
en
by t
wo
equati
ons,
rath
er t
han b
y a
sin
gle
equati
on in x
and y
. T
he
thir
d v
ari
able
is
called
a parameter
, and t
he
two
equati
ons
are
called
parametric equations
of t
he
curv
e.
For
the
curv
es y
ou w
ill gra
ph o
n t
his
page,
the
para
met
er is t and
the
para
met
ric
equati
ons
of e
ach
curv
e are
in t
he
form
x f(t
) and
y g(t
).
Example
Graph the curve associated w
ith the parametric
equationsx 48tandy 64t!16t2.
Choo
se v
alu
es for
tand m
ake
a t
able
show
ing t
he
valu
es o
f all t
hre
e vari
able
s. T
hen
gra
ph t
he
x- and y-v
alu
es.
tx
y
–1
–48
–80
00
00.5
24
28
148
48
296
64
3144
48
4192
0
Graph each curve.
1.x
3t, y
2.x t2#
1, y t3"
112!t
3-3
Answers (Lesson 3-4)
© Glencoe/McGraw-Hill A6 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
97
Ad
van
ced
Math
em
atical C
on
cep
ts
Pra
cti
ce
NAME_____________________________ DATE_______________ PERIOD________
Inverse Functions and Relations
Graph each function and its inverse.
1.ƒ(x
)
(x"
1)3#
12.ƒ(x
)
3 x
#2
Find the inverse of ƒ(x). Then state whether the inverse is also a function.
3.ƒ(x
) "
4x
2#
14.ƒ(x
)
"3
x#"#
1#5.ƒ(x
) !(x"
43)2!
inve
rse
: ƒ!
1(x
) x
3"
1in
ve
rse
: y
3&!2
" xx#!
y &
!1 2!"
1#!#x#
ye
sn
on
o
Graph each equation using the graph of the given parent function.
6.y "
"x###
3#"
1,p
(x) x
27.y
2#
"5
x###
2#,p
(x) x
5
8.Fire Fighting
Air
pla
nes
are
oft
en u
sed t
o dro
p w
ate
r on
for
est
fire
s in
an e
ffor
t to
stop
the
spre
ad o
f th
e fi
re.T
he
tim
e tit
takes
the
wate
r to
tra
vel
fro
m h
eighth
toth
e gro
und c
an b
e der
ived
fro
m t
he
equati
on h
!1 2!gt2
wher
e g
is t
he
acc
eler
ati
ondue
to g
ravit
y (32 fee
t/se
cond
2).
a.
Wri
te a
n e
quati
on t
hat
will giv
e ti
me
as
a funct
ion o
f hei
ght.
t %!
1h 6! &o
r
b.
Suppos
e a p
lane
dro
ps
wate
r fr
om a
hei
ght
of 1
024 fee
t.H
ow m
any s
econ
ds
will it
take
for
the
wate
r to
hit
the
gro
und?
8 s
ec
on
ds
"h #!
4
3-4
© G
len
co
e/M
cG
raw
-Hill
98
Ad
van
ced
Math
em
atical C
on
cep
ts
En
ric
hm
en
t
NAME_____________________________ DATE_______________ PERIOD________
3-4
VE
____
17
28
HA
LF
____
____
229
627
IS
E__
____
31
33
14
ON
E__
____
20
1134
FY
____
36
7R
AT
IO
____
____
__24
16
1910
4E __ 18 L
EF
T__
____
__35
321
8
WH
EN
____
____
13
922
5S
TR
AI
GH
T__
____
____
____
__26
32
30
23
25
12
15
1
1
T14 E
25
I
2
H3
E16 A
4
O17 V
28 E
5
N18 E
29 A
6
L7
Y19 T
26 S
30 R
20 O
8
T9
H21 F
33 S
13 W
15 H
27 F
32 T
31
I
10
I22 E
34 E
11
N23 A
35 L
12 G
24 R
36 F
An Inverse Acrostic
The
puzz
le o
n t
his
page
is c
alled
an a
cros
tic.
To
solv
e th
e puzz
le,
wor
k b
ack
and for
th b
etw
een t
he
clues
and t
he
puzz
le b
ox. Y
ou m
ay
nee
d a
math
dic
tion
ary
to
hel
p w
ith s
ome
of t
he
clues
.
1.I
f a r
elati
on c
onta
ins
the
elem
ent
(e, v), t
hen
the
inver
se o
f th
e re
lati
on m
ust
con
tain
the
elem
ent
( __
, __
).
2.T
he
inver
se o
f th
e fu
nct
ion 2x
is fou
nd b
y
com
puti
ng _
_ of
x.
3.T
he
firs
t le
tter
and t
he
last
tw
o le
tter
s of
the
mea
nin
g o
f th
e sy
mbol
f–1are
__.
4.T
his
is
the
pro
duct
of a n
um
ber
and its
m
ult
iplica
tive
inver
se.
5.If
the
seco
nd c
oord
inate
of th
e in
ver
se o
f (x
, f(x
)) is
y, th
en t
he
firs
t co
ordin
ate
is
read “
__ o
f __
”.
6.T
he
inver
se r
ati
o of
tw
o num
ber
s is
the
__ of
the
reci
pro
cals
of th
e num
ber
s.
7.If)is
a b
inary
oper
ati
on o
n s
et S
and
x)e e)x x for
allx
in S
, th
en a
n iden
tity
el
emen
t fo
r th
e op
erati
on is
__.
8.To
solv
e a m
atr
ix e
quati
on, m
ult
iply
each
sid
e of
the
matr
ix e
quati
on o
n t
he
__ b
y t
he
inver
sem
atr
ix.
9.Tw
o vari
able
s are
inver
sely
pro
por
tion
al __
thei
rpro
duct
is
const
ant.
10.T
he
gra
ph o
f th
e in
ver
se o
f a lin
ear
funct
ion is
a _
_ line.
From President Franklin D. Roosevelt’s inaugural address
during the Great Depression; delivered M
arch 4, 1933.
Answers (Lesson 3-5)
© Glencoe/McGraw-Hill A7 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
101
Ad
van
ced
Math
em
atical C
on
cep
ts
NAME_____________________________ DATE_______________ PERIOD________
Reading Mathematics
The
follow
ing s
elec
tion
giv
es a
def
init
ion o
f a c
onti
nuou
s fu
nct
ion a
s it
mig
ht
be
def
ined
in a
col
lege-
level
math
emati
cs t
extb
ook. N
otic
eth
at
the
wri
ter
beg
ins
by e
xpla
inin
g t
he
not
ati
on t
o be
use
d for
vari
ous
types
of in
terv
als
. It
is
a c
omm
on p
ract
ice
for
colleg
e auth
ors
to e
xpla
in t
hei
r not
ati
on, si
nce
, alt
hou
gh a
gre
at
dea
l of
the
not
ati
on
is s
tandard
, ea
ch a
uth
or u
sually c
hoo
ses
the
not
ati
on h
e or
she
wis
hes
to
use
.
Use the selection above to answer these questions.
1.
What
happen
s to
the
four
ineq
ualiti
es in t
he
firs
t para
gra
ph w
hen
a b?
On
ly t
he
la
st
ine
qu
ality
ca
n b
e s
ati
sfi
ed
. 2.
What
happen
s to
the
four
inte
rvals
in t
he
firs
t para
gra
ph w
hen
a b?
Th
e f
irst
inte
rva
l is
ø a
nd
th
e o
the
rs r
ed
uc
e t
o t
he
po
int a b
.3.
What
math
emati
cal te
rm m
akes
sen
se in t
his
sen
tence
?
If f
(x) is
not
____?__
at x
0, it
is
said
to
be
dis
conti
nuou
s at x
0.
co
nti
nu
ou
s4.
What
not
ati
on is
use
d in t
he
sele
ctio
n t
o ex
pre
ss t
he
fact
that
a n
um
ber
xis
co
nta
ined
in t
he
inte
rval I?
x�I
5.
In t
he
space
at
the
right,
sket
ch t
he
gra
ph
of t
he
funct
ion f(x) def
ined
as
follow
s:
f(x)
$if x
�'0,
(
1, if x �
', 1)
6.
Is t
he
funct
ion g
iven
in E
xer
cise
5 c
onti
nuou
s on
the
inte
rval [0, 1]?
If not
, w
her
e is
the
funct
ion d
isco
nti
nuou
s?
1 ! 2
1 ! 21 ! 2
En
ric
hm
en
t3-5
Throughout this book, the set
S, called the domain of definition of a function, will usually be
an interval. An interval is a set of numbers satisfying one of the four inequalities
a'
x'
b,
a(
x'
b,a'
x(
b,or
a(
x(
b.In these inequalities,
a(
b.The usual notations for the
intervals corresponding to the four inequalities are, respectively, (a,b), [
a, b), (
a, b], and [
a, b].
An interval of the form (
a,b) is called o
pen, an interval of the form
[a, b)or(a, b] is called
half-o
penor
half-c
losed, and an interval of the form
[a,b] is called
clo
sed.
Suppose
Iis an interval that is either open, closed, or half-open. Suppose
ƒ(x)is a function
defined on
Iandx
0 is a point in
I. We say that the functionf(x
) is continuous at the point
x0if
the quantity |
ƒ(x)"
ƒ(x0)|
becomes small as x
�Iapproaches
x0.
No
; it
is d
isc
on
tin
uo
us a
t x
.1 ! 2
© G
len
co
e/M
cG
raw
-Hill
100
Ad
van
ced
Math
em
atical C
on
cep
ts
Continuity and End Behavior
Determine whether each function is continuous at the given x-value. Justify
your answer using the continuity test.
1.y !32 x
2!
; x "
12.y !x
2#
2x#
4!
; x
1
Ye
s;
the
fu
nc
tio
n is d
efi
ne
d
Ye
s; th
e f
un
cti
on
is
at x !
1; y
ap
pro
ac
he
s !2 3!
as
de
fin
ed
at x
1; y
xa
pp
roa
ch
es!
1 f
rom
bo
th
ap
pro
ac
he
s 3
as x
ap
pro
ac
he
s
sid
es; ƒ(!
1)
!2 3!.
1 f
rom
bo
th s
ide
s; ƒ(1
)
3.
3.y x
3"
2x#
2; x
14.y !x x" #
2 4!
; x "
4
Ye
s;
the
fu
nc
tio
n is d
efi
ne
d
No
; th
e f
un
cti
on
is
at x
1; y a
pp
roa
ch
es 1
as
un
de
fin
ed
at x !
4.
xa
pp
roa
ch
es 1
fro
m b
oth
sid
es;
ƒ(1
)
1.
Describe the end behavior of each function.
5.y
2x
5"
4x
6.y "
2x
6#
4x
4"
2x#
1
y→∞
as x→∞
,y→!∞
as x→∞
,y→!∞
as x→!∞
y→!∞
as x→!∞
7.y x
4"
2x
3#x
8.y "
4x
3#
5
y→∞
as x→∞
,y→!∞
as x→∞
,y→∞
as x→!∞
y→∞
as x→!∞
Given the graph of the function, determine the interval(s) for which the function
is increasing and the interval(s) for which the function is decreasing.
9.
inc
rea
sin
g f
or x#!
1 a
nd
x% 1
; d
ec
rea
sin
g f
or!
1#x#
1
10.Electronics
Ohm
’s L
aw
giv
es t
he
rela
tion
ship
bet
wee
n r
esis
tance
R,voltage
E,and curren
t Iin a circuit as R
!E I!.If the voltage remains constant but the
curren
t keeps increasing in the circuit, what happen
s to the resistance?
Re
sis
tan
ce
de
cre
ase
s a
nd
ap
pro
ac
he
s z
ero
.
Pra
cti
ce
NAME_____________________________ DATE_______________ PERIOD________
3-5
Answers (Lesson 3-6)
© Glencoe/McGraw-Hill A8 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
104
Ad
van
ced
Math
em
atical C
on
cep
ts
"Unreal" Equations
There are som
e equation
s that cannot be graphed on the
real-number coordinate system
. One example is the equation
x2– 2x"
2y2"
8y"
14
0. Com
pleting the squares in xand y
gives the equation
(x – 1)2"
2(y"
2)2
–5.
For any real numbers, xand y, the values of (x – 1)2and 2(y"
2)2
are non
negative. So, their sum cannot be –5. Thus, no real values of
xand ysatisfy the equation
; on
ly imaginary values can be solution
s.
Determine whether each equation can be graphed on the
real-number plane. Write yes or no.
1.(x"
3)2"
(y#
2)2 #4
2.x2#
3x"y2"
4y #7
no
no
3.(x"
2)2"y2#
6y"
8
04.x2"
16
0
ye
sn
o5.x4"
4y2"
4
06.x2"
4y2"
4xy"
16
0
no
no
In Exercises 7 and 8, for what values of k :
a. will the solutions of the equation be imaginary?
b. will the graph be a point?
c. will the graph be a curve?
d. Choose a value of k for which the graph is a curve and sketch
the curve on the axes provided.
7.x2#
4x"y2"
8y"k
08.x2"
4x"y2#
6y#k
0
a. k
> 2
0; b
. k
= 2
0;
a. k
< –
13; b
. k
= –
13;
c. k
< 2
0;
c. k
> –
13;
d.
d.
Enrichment
NAME_____________________________ DATE_______________ PERIOD________
3-6
© G
len
co
e/M
cG
raw
-Hill
103
Ad
van
ced
Math
em
atical C
on
cep
ts
Practice
NAME_____________________________ DATE_______________ PERIOD________
Critical Points and Extrema
Locate the extrema for the graph of y ƒ(x). Name and classify the
extrema of the function.
1.
2.
rela
tive
ma
xim
um
: (!
2,
1)
ab
so
lute
min
imu
m:
(!2
, 0
)re
lati
ve
min
imu
m:
(1,!
2)
ab
so
lute
min
imu
m:
(!6
,!
6)
3.
4.
ab
so
lute
ma
xim
um
: (!
1,
3)
rela
tive
ma
xim
um
: (0
, 2
)re
lati
ve
min
imu
m:
(2,!
2)
Determine whether the given critical point is the location of a maximum, a
minimum, or a point of inflection.
5.y x2#6x"1, x 3
6.y x2#2x#6, x 1
7.y x4"3x2#5, x 0
min
imu
mm
inim
um
min
imu
m
8.y x5#2x3#2x2, x 0
9.y x3"x2#x, x #1
10.y 2x3"4, x 0
ma
xim
um
ma
xim
um
po
int
of
infl
ec
tio
n
11.Physics
Suppose that during an experim
ent you
launch a toy rocket straight upward from a height
of 6 inches with an initial velocity of 32 feet per
second. The height at any tim
e tcan be modeled by
the function s(t) #16t2"
32t"
0.5 wheres(t) is
measured in feet and tis m
easured in secon
ds.
Graph the function to find the maxim
um height
obtained by the rocket before it begins to fall.
16
.5 f
t
3-6
Answers (Lesson 3-7)
© Glencoe/McGraw-Hill A9 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
107
Ad
van
ced
Math
em
atical C
on
cep
ts
Enrichment
NAME_____________________________ DATE_______________ PERIOD________
Slant Asymptotes
The graph of y ax"b, where a$
0, is called a slant asymptote of
y f(x) if the graph of f(x)comes closer and closer to the line as
x�∞or x
�–∞.
Forf(x)
3x"
4"
,y
3x"
4 is a slant asymptote because
f(x) #
(3x"
4)
, and
�0 as x
�∞or x
�–∞.
Example
Find the slant asymptote off(x)
.
–2
18
15
Use synthetic division.
–2–12
16
3
y
x"
6"
Since
�0 as x
�∞
or x
�–∞,
y x"
6 is a slant asymptote.
Use synthetic division to find the slant asymptote for each of the following.
1.y
y
8x#
44
2.y
y x"
5
3.y
y x"
1
4.y
y ax"b"ad
5.y
y ax"b#ad
ax2"bx"c
!!
x"d
ax2"bx"c
!!
x#d
x2#
2x#
18
!!
x#
3
x2"
3x#
15
!!
x#
2
8x2#
4x"
11!!
x"
5
3! x"
2
3! x" 2
x2"
8x"
15
!!
x"
2
x2"8x"15
##
x"2
2 ! x
2 ! x
2 ! x
3-7
© G
len
co
e/M
cG
raw
-Hill
106
Ad
van
ced
Math
em
atical C
on
cep
ts
Graphs of Rational Functions
Determine the equations of the vertical and horizontal asymptotes, if any, of
each function.
1.ƒ(x) ! x2
4 "1!
2.ƒ(x) !2 xx ""
11!
3.g(x) ! (x"
x 1" )(x3 #
2)
!
y
0x !
1, y
2x !
1, x
2, y
0
Use the parent graph ƒ(x)
!1 x!to graph each equation. Describe the
transformation(s) that have taken place. Identify the new locations
of the asymptotes.
4.y ! x"3
1!#
25.y #! x#4
3!"
3
exp
an
de
d v
ert
ica
lly b
y a
refl
ec
ted
ove
r x-a
xis
;fa
cto
r o
f 3
; tr
an
sla
ted
le
fte
xp
an
de
d v
ert
ica
lly b
y a
1 u
nit
an
d d
ow
n 2
un
its;
fac
tor
of
4;
tra
nsla
ted
rig
ht
x !
1, y !
23 u
nit
s a
nd
up
3 u
nit
s; x
3, y
3
Determine the slant asymptotes of each equation.
6.y !5x2# x#10 2x"
1!
7.y !x x
2
"#
1x!
y
5x
y x!
2
8.Graph the function y !x2
x""
x1#
6!
.
9.Physics
The illumination
Ifrom
a light source is given
by the
form
ula I
! dk 2!, where kis a con
stant and d
is distance. A
s the
distance from the light source dou
bles, how
does the illumination
change?
It d
ec
rea
se
s b
y o
ne
fo
urt
h.
Practice
NAME_____________________________ DATE_______________ PERIOD________
3-7
Answers (Lesson 3-8)
© Glencoe/McGraw-Hill A10 Advanced Mathematical Concepts
© G
len
co
e/M
cG
raw
-Hill
110
Ad
van
ced
Math
em
atical C
on
cep
ts
Reading Mathematics: Interpreting
Conditional Statements
The conditional statemen
t below
is written
in “if-then
”form
.It has
the form
p →
qwhere pis the hypothesis and q
is the consequen
t.
If a m
atrix A
has a determinant of 0,then A!1does not exist.
It is im
portant to recognize that a con
ditional statemen
t need not
appear in “if-then
”form
.For example,the statemen
t
Any point that lies in Quadrant I has a positive x-coordinate.
can be rewritten
as
If the point P(x,y) lies in Quadrant I,then x
is positive.
Notice that Plying in Quadrant I is a sufficientcondition for its
x-coordinate to be positive.
Another way to express this is to say
that Plying in Quadrant I guarantees
that its x-coordinate is
positive.
On the other hand,we can also say that xbeing positive is
a necessary
condition for P
to lie in Quadrant I.In other words,P
does not lie in Quadrant I if xis not positive.
To change an English statemen
t into “if-then
”form
requires that
you
understand the meaning and syntax of the English statemen
t.Study each of the follow
ing equivalent ways of expressing p→q.
• If pthen
q• p
implies q
• p
only if q
• only if q,p
•pis a sufficient condition for q
• not p
unless q
• q
is a necessary con
dition for p.
Rewrite each of the following statements in “if-then” form.
1.A consistent system
of equations has at least one solution.
If a
syste
m o
f e
qu
ati
on
s is c
on
sis
ten
t, t
he
n t
he
syste
m h
as a
t le
ast
on
e s
olu
tio
n.
2.When
the region formed by the inequalities in a linear
programming application
is unbou
nded,an optimal solution
for the problem m
ay not exist.
If t
he
re
gio
n f
orm
ed
by t
he
in
eq
ua
liti
es in
a lin
ea
r p
rog
ram
min
g a
pp
lic
ati
on
is u
nb
ou
nd
ed
, th
en
an
o
pti
ma
l so
luti
on
fo
r th
e p
rob
lem
ma
y n
ot
exis
t.3.Functions whose graphs are symmetric with respect to the y-axis are
called even functions.
If t
he
gra
ph
of
a f
un
cti
on
is s
ym
me
tric
wit
h
resp
ec
t to
th
e y
-axis
, th
en
th
e f
un
cti
on
is e
ve
n.
4.In order for a decim
al number d
to be odd,it is sufficient that den
d in the digit 7.
If a
de
cim
al n
um
be
r d
en
ds in
th
e d
igit
7, th
en
dis
od
d.
Enrichment
NAME_____________________________ DATE_______________ PERIOD________
3-8
© G
len
co
e/M
cG
raw
-Hill
109
Ad
van
ced
Math
em
atical C
on
cep
ts
Practice
NAME_____________________________ DATE_______________ PERIOD________
Direct, Inverse, and Joint Variation
Write a statement of variation relating the variables of each equation.
Then name the constant of variation.
1.#
!x y
2 !
32.E IR
yva
rie
s d
ire
ctl
y a
s t
he
Eva
rie
s j
oin
tly a
s
sq
ua
re o
f x;!
!1 3!I
an
d R
; 1
3.y
2x
4.d
6t2
yva
rie
s d
ire
ctl
y a
s x
; 2
dva
rie
s d
ire
ctl
y a
s
the
sq
ua
re o
f t;
6
Find the constant of variation for each relation and use it to write an
equation for each statement. Then solve the equation.
5.Suppose yvaries directly as xand y
35 when
x
5. Find ywhen
x
7.
7; y
7x;4
9
6.If yvaries directly as the cube of xand y
3 when
x
2, find xwhen
y
24.
!3 8!; y
!3 8!x
3;
4
7.If yvaries inversely as xand y
3 when
x
25, find xwhen
y
10.
75
; y
!7 x5 !;7
.5
8.Suppose yvaries jointly as xand z, and y
64 when
x
4 and z
8.
Find ywhen
x
7 and z
11.
2; y
2xz;1
54
9.Suppose Vvaries jointly as hand the square of r,and V
45
�when
r
3 and
h
5. Findrwhen
V
175
�and h
7.
�; V
�r2h
;5
10.If yvaries directly as xand inversely as the square of z, and y #5 when
x
10 and
z
2, find ywhen
x
5 and z
5.
!2; y !
2! zx
2!; !
0.4
11.Finances
Enrique deposited $200.00 into a savings accou
nt. The simple
interest Ion
his accou
nt varies jointly as the time tin years and the principalP.
After one quarter (three mon
ths), the interest on Enrique’s accou
nt is $2.75.
Write an equation
relating interest, principal, and tim
e. Find the constant of
variation
. Then
find the interest after three quarters.
I kPt; 0
.05
5;
$8
.25
3-8
© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts
Page 111
1. C
2. B
3. D
4. C
5. B
6. D
7. A
8. D
9. A
10. D
Page 112
11. B
12. B
13. C
14. C
15. A
16. D
17. A
18. B
19. A
20. D
Bonus: D
Page 113
1. B
2. A
3. B
4. A
5. C
6. C
7. B
8. A
9. B
10. D
Page 114
11. C
12. A
13. C
14. C
15. C
16. A
17. D
18. B
19. A
20. B
Bonus: B
Chapter 3 Answer KeyForm 1A Form 1B
© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts
Chapter 3 Answer Key
Page 115
1. B
2. C
3. C
4. C
5. C
6. C
7. C
8. A
9. B
10. D
Page 116
11. A
12. D
13. A
14. C
15. C
16. C
17. B
18. D
19. A
20. B
Bonus: A
Page 117
1. y x, y !x,the origin
2. x-axis
3. odd
4.
5.
6.
7. x$ 8
8. ƒ!1(x) !1
2!
xx! ; yes
9. ƒ!1(x) & x! "! 4! ;no
10. yes
Page 118
11. discontinuous jump
12. discontinuous infinite
13. x→ ∞, y→!∞, x→!∞, y→!∞
14. rel. max. at (0, 2);abs. min. at (&1.22, !0.25)
15. point of inflectionat (1, 1)
16. horizontal at y 0
17. y 3x " 1
18.
19. y 16
20. k !21!
Bonus:
Form 1C Form 2A
reflected over the x-axis, expandedvertically by factorof 2, translated up 4 units
x'!3 or
vertical at x 0, 3;
© Glencoe/McGraw-Hill A13 Advanced Mathematical Concepts
Page 119
1. y x, y !x,the origin
2. origin
3. even
4.
5.
6.
7. !3' x ' 7
8.
9.
10. no
Page 120
11. continuous
12.
13.
14.
15.
16.
17. y 2x " 1
18.
19. y 27
20. k !43!�
Bonus: k 6
Page 121
1. y-axis
2. origin
3. even
4.
5.
6.
7. !4 ' x' 12
8. ƒ!1(x) & x!; no
9.
10. yes
Page 122
11.
12. continuous
13.
14. abs. max. at (0, 0)
15. point of inflection
at (0, 0)
16. vertical at x &5;
horizontal at
y 0
17. y x
18.
19. y 32
20. y 15
Bonus: k 0
Chapter 3 Answer KeyForm 2B Form 2C
translated right3 units,compressedvertically by afactor of 0.5
ƒ!1(x) 3x! "! 4!;
yes
ƒ!1(x) 1& x! ; no
discontinuous;point
x→ ∞, y→!∞,x→ ∞, y→∞
rel. max. at (0, 0);abs. min. at(&1.22, !2.25)
point of inflection
at (!2, 1)
vertical at x 0,1; horizontal at y 0
translated right1 unit
ƒ!1(x) 3
x!"! 1!;yes
discontinuous;jump
x→ ∞, y→ ∞, x→!∞, y→∞
© Glencoe/McGraw-Hill A14 Advanced Mathematical Concepts
Chapter 3 Answer Key
CHAPTER 3 SCORING RUBRIC
Level Specific Criteria
3 Superior • Shows thorough understanding of the concepts parentgraph, transformation asymptote, hole in graph, critical point, maximum, minimum, and point of inflection.
• Uses appropriate strategies to solve problems and transform graphs.
• Computations are correct.
•Written explanations are exemplary.
• Graphs are accurate and appropriate.
• Goes beyond requirements of some or all problems.
2 Satisfactory, • Shows understanding of the concepts parent graph, with Minor transformation, asymptote, hole in graph, critical point,Flaws maximum, minimum, and point of inflection.
• Uses appropriate strategies to solve problems andtransform graphs.
• Computations are mostly correct.
•Written explanations are effective.
• Graphs are mostly accurate and appropriate.
• Satisfies all requirements of problems.
1 Nearly • Shows understanding of most of the concepts parentSatisfactory, graph, transformation, asymptote, hole in graph, criticalwith Serious point, maximum, minimum, and point of inflection.
Flaws • May not use appropriate strategies to solve problems andtransform graphs.
• Computations are mostly correct.
•Written explanations are satisfactory.
• Graphs are mostly accurate and appropriate.
• Satisfies most requirements of problems.
0 Unsatisfactory • Shows little or no understanding of the concepts parentgraph, transformation, asymptote, hole in graph, critical point, maximum, minimum, and point of inflection.
• May not use appropriate strategies to solve problems and transform graphs.
• Computations are incorrect.
•Written explanations are not satisfactory.
• Graphs are not accurate or appropriate.
• Does not satisfy requirements of problems.
© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts
Page 123
1a.
1. parent graph 2. transformation
3. transformation 4. transformation
The parent graph y x2 is reflectedover the x-axis and translated 5 unitsright and 2 units down.
1b.
1. parent graph 2. transformation
3. transformation
The parent graph y x3 is translated 2 units left and 1 unit up.
1c.
Since x ! 4 is a common factor of the numerator and the denominator,the graph has point discontinuity at x 4. Because y increases ordecreases without bound close to x !3 and x 0, there are verticalasymptotes at x !3 and x 0.
1d. ƒ(x) !(x !
x2"
)(x1" 1)!
ƒ(x) !x!
12!, except at
x !1, where it is undefined.
ƒ(x) !x!
12! is undefined at x 2.
It approaches !∞ to the left of x 2 and "∞ to the right.
2. #!1, !16303!$ is a maximum. (0, 2) is a
point of inflection. #2, ! !1145!$ is a
minimum. The value x 3 is not acritical point because a line drawntangent to the graph at this point isneither horizontal nor vertical. Thegraph is increasing at x 2.99 and at x 3.01.
3a. ƒ(x) total income, x number of weeks ƒ(x) (6.00 " 0.10x)(100 ! x)
!0.10x2" 4x " 600
Examining the graph of the functionreveals a maximum at x 20. ƒ( 20) 640, ƒ( 19.9) 639.999, andƒ( 20.1) 639.999, so ƒ( 19.9) ( ƒ( 20)and ƒ( 20) ( ƒ( 20.1), which showsthat the function has a maximum of640 when x 20.
3b. She should sell immediately becausethe income function would always bedecreasing.
Chapter 3 Answer KeyOpen-Ended Assessment
© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts
Mid-Chapter TestPage 124
1. x-axis
2. y x, y !x, origin
3. neither
4.
5.
6.
7. x '!2 or x $ 10
8.
9. ƒ!1(x) !!1x!; yes
10. no
Quiz APage 125
1. none of these
2. y-axis
3. odd
4.
5.
Quiz BPage 125
1.
2. {x|x( !7 or x) 1}
3. ƒ!1(x) !x "3
6! ; yes
4.
5. y x
Quiz CPage 126
1. discontinuous; jump
2. discontinuous; infinite
3. x →∞, y →∞, x →!∞, y →∞
4. rel. max. (!1, 4); rel. min. (1, !4)
5. min. at (2, !4.27)
Quiz DPage 126
1. vertical at x 2 and 3, horizontal at y 1
2. y x! 2
3.
4. 1.5; y 1.5x; 21
5. 2; y 2xz3; !500
Chapter 3 Answer Key
translated up 1 unit,expanded horizontallyby a factor of 2
ƒ!1(x) 3x! !! 1!; yes
g(x) is the reflection of ƒ(x) over
the x-axis and is expanded
vertically by a factor of 4
translated right 3 units,expanded vertically by afactor of 2; verticalasymptote at x 3,horizontal asymptoteunchanged at y 0
© Glencoe/McGraw-Hill A17 Advanced Mathematical Concepts
Page 127
1. C
2. B
3. A
4. B
5. C
6. A
7. D
8. A
9. E
Page 128
10. B
11. D
12. E
13. B
14. C
15. A
16. D
17. A
18. A
19. 8
20. !1
Page 129
1. D {!2, 0, 2}; R {2}; yes
2. !x3!
3. 2x ! y" 7 0
4.
5. (2, 3)
6. (2, !1, 3)
7. one
8. 29
9. origin
10.
11. !1( x( 5
12. ƒ!1(x) !x3! " 2
13. jump
14. y 1
15. 15
Chapter 3 Answer KeySAT/ACT Practice Cumulative Review
expanded vertically by afactor of 2, translated right1 unit