Symmetry and Coordinate Graphs - Vista Peak Prep...

© Glencoe/McGraw-Hill 87 Advanced Mathematical Concepts

Study Guide

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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.


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

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x

f x( )

O

(3, 3)

(4, 2)(1, 1)


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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.


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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.


Practice

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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.

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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.


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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}.

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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

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Enrichment

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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

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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.

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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


Practice

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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?

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Enrichment

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__ __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.


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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).



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

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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.


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.

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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


Practice

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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.

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"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

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Study Guide

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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).

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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


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


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


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


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


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


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


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


Chapter 3 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________


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


Chapter

3


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)


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)


3


Chapter 3 Test, Form 1C

NAME _____________________________ DATE _______________ PERIOD ________


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


Chapter

3


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)


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)


3


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


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)


3


Chapter 3 Test, Form 2B

NAME _____________________________ DATE _______________ PERIOD ________




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. __________________



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




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)


3


Chapter 3 Test, Form 2C

NAME _____________________________ DATE _______________ PERIOD ________




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. __________________



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




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)


3


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


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. __________________



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)


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 ________


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 ________


Chapter

3

Chapter

3


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


Chapter 3 SAT and ACT Practice (continued)


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

}


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


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)


© 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

.



© 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.



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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

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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



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raw

-Hill

97

Ad

van

ced

Math

em

atical C

on

cep

ts

Pra

cti

ce

NAME_____________________________ DATE_______________ PERIOD________


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

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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.



© G

len

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e/M

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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

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100

Ad

van

ced

Math

em

atical C

on

cep

ts


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



© G

len

co

e/M

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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

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Practice

NAME_____________________________ DATE_______________ PERIOD________


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

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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

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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

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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

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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


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


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;


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→∞



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.


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


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


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


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

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