chapter 13: advanced functions and relations

T H E M E : Astronomy

Have you ever tried counting the stars at night? On a clear night, far awayfrom bright city lights, you can see about 3000 stars using only your eyes.

Today, astronomers use computers, telescopes, and satellites. Through color-spectrum analysis, they explore the history of our galaxy and solar system.

• Astronauts (page 571) are pilots and scientists, who travel in space. Ona space shuttle, there are pilots, mission specialists, and payloadspecialists. They oversee and conduct experiments.

• Astronomers (page 589) use physics and mathematics to study theuniverse through observation and calculation. The knowledge gainedthrough the science of astronomy helps inrelated fields of navigation and spaceflight. Astronomers work withengineers to design and launchspace probes and satellites togather data and transmit itback to Earth.

558 mathmatters3.com/chapter_theme

Advanced Functionsand Relations

CH

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1313

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Chapter 13 Advanced Functions and Relations

Use the table for Questions 1–4.

1. To the nearest million square miles, what is the surface area ofthe Earth? Assume the planet’s shape is spherical.

2. Using the orbital speed and length of year in Earth days,calculate the length of Venus’ orbit in miles.

3. An astronomer is 58 years old in Earth years. What is theastronomer’s age in Martian years?

4. Find the circumference of Uranus to the nearest mile.

CHAPTER INVESTIGATIONHow far is the Earth from the Sun? Because theEarth’s orbit is elliptical, its distance from the Sun varies from 91.4 million miles to 94.5 million miles. The closest orbital point is called the perihelion. The farthest orbital point is called the aphelion. Astronomers have calculated each planet’s aphelion and perihelion.

Working TogetherVast distances in space are difficult to imagine.Reducing these distances to a familiar scale canmake them easier to visualize. Research theaphelion and perihelion for the nine planets of the Solar System. Then choose a location, such as your home or school, to represent the Sun. Using a system of maps, plot the aphelion and perihelion for each planet. Use the Chapter Investigation icons to guide your group.

Data Activity: The Solar System

559

197,000,000 mi

423,226,944 mi

30 years old

99,742 mi

The Solar System

Planet

Mercury

Venus

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

Pluto

87.97 Earth days

224.7 Earth days

365.26 Earth days

1.88 Earth years

11.86 Earth years

29.46 Earth years

84.01 Earth years

164.79 Earth years

248.54 Earth years

Length of year

0.055

0.81

1

0.11

318

95.18

14.5

17.14

0.0022

Mass inEarth masses

29.8 mi/sec

21.8 mi/sec

18.5 mi/sec

15 mi/sec

8.1 mi/sec

6 mi/sec

4.2 mi/sec

3.4 mi/sec

2.9 mi/sec

Averageorbital speed

3031 mi

7521 mi

7926 mi

4217 mi

88,850 mi

74,901 mi

31,765 mi

30,777 mi

1429 mi

Diameter

The skills on these two pages are ones you have already learned. Review theexamples and complete the exercises. For additional practice on these and moreprerequisite skills, see pages 654–661.

CIRCLES

Because you will be doing a lot of work with circles in this chapter, it may behelpful to review some of the basics.

Example What is the diameter and the circumference of this circle?

AB is a radius (r) of the circle.RS is a diameter (d) of the circle.A is the center point of the circle.

d � 2r C � 2�r� 2(3) � 2(3.14)(3)� 6 cm � 18.84 cm

Find the diameter and circumference for each circle. Use 3.14 for �. Roundanswers to the nearest hundredth if necessary.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.2.97 km

8.7 mm37.2 dm

0.8 ft9.6 in.

4.2 cm

17 km

4 mi

3.9 dm

8 ft1.5 in.7 cm


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13 Are You Ready?Refresh Your Math Skills for Chapter 13

d � 14 cm;C � 43.96 cm

d � 3 in., C � 9.42 in.

d � 16 ft;C � 50.24 ft

d � 34 km;C � 106.76 km

d � 8 mi;C � 25.12 mi

d � 7.8 dm;C � 24.49 dm

d � 8.4 cm;C � 26.38 cm

d � 19.2 in.;C � 60.29 in.

d � 1.6 ft;C � 5.02 ft

d � 5.94 km;C � 18.65 km

d � 17.4 mm;C � 54.64 mm

d � 74.4 dm;C � 233.62 dm

3 cm

R

A

S

B

561Chapter 13 Are You Ready?

SOLVE PROPORTIONS

Example Find x: �192� � �

3x0�

Use cross-multiplication: �192� � �

3x0�

9 � 30 � 12 � x270 � 12x22.5 � x

Find the value of x in each proportion. Round answers to the nearest tenth ifnecessary.

13. �142� � �

2x8� 14. �

96

� � �3x0� 15. �

45

� � �10

x0

�

16. �85

� � �4x4� 17. �

986� � �

2x4� 18. �

104

8� � �

6x

�

19. �398� � �

1x4� 20. �

250� � �

7x

� 21. �137� � �

2x2�

22. �4x

� � �15

56� 23. �

11

93� � �

3x1� 24. �

78

16� � �

5x0�

SOLVE SYSTEMS OF EQUATIONS BY GRAPHING

Example Solve: 2x � y � 1

y � x � �5

To solve, graph both equations on the same coordinate plane. The point at which the graphs intersect is the solution.

The solution is (2, �3).

Solve each system of equations by graphing.

25. 3x � 2y � �11 26. x � y � 2

�2x � 3y � 9 3y � x � 2

27. x � 2y � �2 28. x � 6y � �1

2y � 2 � x 4y � x � 1

29. 2x � 2 � y 30. 4y � 3x � 5

3y � 5x � 3 x � 3 � 2y

�5

5

5

�5 x

y

9.3

27.5

59.1

1.1 21.2

28

2

20 125

162

124.7

41.3

(�3, 1)

(4, 2)

(1, 2)(�3, �4)

infinitely manysolutions

��15

�, �15

��

Work with a partner. You will need a compass or geometry software.

1. Draw a circle on a coordinate grid with a radius of 5 units.

2. Complete the table below by estimating the missing y-coordinates for pointson the circle. Then find the values of the expressions in the third and fourthcolumns. Note that there will be two different y-coordinates for each x-coordinate.

3. Explain the patterns that you see in the table.

BUILD UNDERSTANDING

You can substitute into the distance formula to find the equation for a circle withits center located at any coordinate point (h, k).

d � �(x2 ��x1)2 �� (y2 �� y1)2�

r � �(x � h�)2 � (y� � k)2�

r 2 � (x � h)2 � (y � k)2

The standard equation of a circle is

(x � h)2 � (y � k)2 � r 2, r � 0.

If h � 0 and k � 0, then the standard equation simplifies to x 2 � y 2 � r 2.

E x a m p l e 1

Write an equation for a circle with radius 6 units and center (0, 0).

SolutionBecause the center is at the origin, substitute in the equation x 2 � y 2 � r 2.

x 2 � y 2 � 62 or

x 2 � y 2 � 36

13-1 The Standard Equation of a CircleGoals ■ Write equations for circles.

Applications Astronomy, Sports, Architecture

y

x(0, 0)

r

(x, y)

x y x � y � x � y

3300

1.81.8

2 2 2 2

Check students’ work.

Estimations will vary. Samples given.

Answers will vary.

d � r, (x1, y1) � (h, k), (x2, y2) � (x, y)

Square both sides.

4 25 5�4 25 5

5 25 5�5 25 5

4.7 25.33 5.03�4.7 25.33 5.03

Chapter 13 Advanced Functions and Relations562

E x a m p l e 2

ASTRONOMY An astronomer is creating a computer model of a moon by entering the equation for a circle with a radius of 4 units and with the center located at point (3, �2). Write the equation.

SolutionSubstitute into the standard form for the equation of a circle.

(x � h)2 � (y � k)2 � r 2

(x � 3)2 � (y � (�2))2 � 42

(x � 3)2 � (y � 2)2 � 42 or

(x � 3)2 � (y � 2)2 � 16

E x a m p l e 3

GRAPHING Find the radius and center of thecircle x 2 � y 2 � 9. Then graph the circle using agraphing calculator.

SolutionBecause the equation is of the form x 2 � y 2 � r 2, the center is at the origin.

x 2 � y 2 � r 2

x 2 � y 2 � 9

r 2 � 9

r � 3

To graph a circle using a graphing calculator,rewrite the equation in terms of y.

x 2 � y 2 � 9

y 2 � 9 � x 2

y � � �9 � x 2�

For most graphing calculators, you must enter separate formulas for the upper and lower portions of the circle. At the Y� screen, enter the functions Y1 � �9 � x 2� and Y2 � ��9 � x 2�.

Use the ZSquare tool from the ZOOM menu to adjust the display so that each pixel represents a square. The graph is a circle with radius 3 and center at the origin.

You can also find the radius and center of a circle using the standard form for theequation of a circle.

Lesson 13-1 The Standard Equation of a Circle 563

4

2

�2

�4

2 4�2�4

y

x

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E x a m p l e 4

Find the radius and center of the circle (x � 5)2 � (y � 4)2 � 18.

SolutionFind what you would substitute into the standard form for the equation of a circle to get the given equation.

(x � h)2 � (y � k)2 � r 2

(x � 5)2 � (y � (�4))2 � (�18�)2

You would substitute 5 for h, � 4 for k, and �18� for r. So, the center (h, k) is (5, �4). The radius is �18� or 3�2�.

TRY THESE EXERCISES

Write an equation for each circle.

1. radius 7 2. radius 10 3.center (0, 0) center (�3, 0)

Find the radius and center for each circle.

4. x 2 � y 2 � 100 5. x 2 � y 2 � 11

6. (x � 4)2 � (y � 2)2 � 49 7. (x � 8)2 � (y � 4)2 � 13

8. WRITING MATH Does the equation for a circle describe a function? Explain your reasoning.

PRACTICE EXERCISES


9. radius 12 10. radius 9center (0, 0) center (4, �6)

11. radius 12 12. radius 11center (�9, �4) center (0, 4)

13. DESIGN The position of a circular knob on a design for a DVD player isshown on the graph below. Find the equation of the circle.

14. SCIENCE In a model, two metal balls are attached to the endpointsof a metal rod. The rod is attached to a machine at its center point. Themachine spins the rod so that the balls move in a circle modeled by theequation x 2 � y 2 � 144. Find the length of the rod.

2

�2

�4

y

x2�2 4 6 8

4

�4

�8

y

x4�4�8�16

x2 � y2 � 49

(x � 3)2 � y2 � 100

(x � 8)2 � y2 � 25

10, (0, 0) �11�, (0, 0)

�13�, (�8, 4)7, (4, 2)

x2 � y2 � 144 (x � 4)2 � (y � 6)2 � 81

(x � 9)2 � (y � 4)2 � 144 x2 � (y � 4)2 � 121

24 units

(x � 5)2 � (y � 1)2 � 4

No, the graph does not pass the vertical line test.

PRACTICE EXERCISES • For Extra Practice, see page 703.


15. SPORTS A circular target is set up for hang glider landings.Write an equation to model a circle with a diameter of 5.2 meters and a center of (0, 0).


16. x 2 � y 2 � 121

17. x 2 � y 2 � 15

18. (x � 1)2 � (y � 3)2 � 9

19. (x � 4)2 � (y � 2)2 � 14

20. (x � 5)2 � (y � 12)2 � 17

21. (x � 12)2 � (y � 6)2 � 20

22. x 2 � (y � 4)2 � 17

23. (x � 9)2 � (x � 1)2 � 400

24. (x � 3)2 � y 2 � 50

25. (x � 1)2 � (y � 9)2 � 81

26. ARCHITECTURE In the landscaping plan for a museum lawn, four circularpaths are designed to intersect at the origin as shown below. Write theequation for each circle.

EXTENDED PRACTICE EXERCISES

Write two equations for each circle, given the coordinatesof the endpoints of the radius.

27. (0, 0), (9, 0)

28. (2, �3), (4, �3)

Find the radius and center point for each circle.

29. x 2 � y 2 � 4x � 2y � 4

30. x 2 � y 2 � 16x

MIXED REVIEW EXERCISES

For each function, name the coordinates of the vertex of the graph. (Lesson 12-1)

31. y � 3x 2 32. y � x 2 � 4

33. y � �2x 2 � 3 34. y � �x 2 � 2

35. y � �4x 2 � 6 36. y � x 2 � 5

Find the slope of the line containing the given points. (Lesson 6-1)

37. A(�3, 4), B(6, �2) 38. C(3, 8), D(�2, �5) 39. E(�4, �5), F(5, 4)

40. G(7, �3), H(2, 5) 41. J(�2, 6), K(�2, �5) 42. L(4, �3), M(�4, 6)

43. N(5, 3), P(�4, �7) 44. Q(�5, 3), R(2, 3) 45. S(3, 0), T(�3, 3)

46. R(�4, �1), T(�3, �3) 47. P(�3, 3), W(1, 3) 48. J(�2, 1), K(�2, 3)

49. C(2, 3), D(9, 7) 50. A(5, 7), B(�2, �3) 51. M(�3, 6), N(2, 4)

52. T(�3, �4), W(5, �1) 53. R(2, �1), S(5, �3) 54. Y(2, 6), Z(�1, 3)

5

3

�3

�5

y

x3�3�5 5

x2 � y2 � 6.76

11, (0, 0)

�15�, (0, 0)

3, (1, 3)

�14�, (�4, 2)

�17�, (�5, �12)

2�5�, (12, �6)

�17�, (0, 4)

20, (9, �1)

5�2�, (�3, 0)

9, (�1, 9)

x2 � (y � 3)2 � 9x2 � (y � 3)2 � 9(x � 3)2 � y2 � 9(x � 3)2 � y2 � 9

For 27–28, see additional answers.

3, (2, 1)

8, (8, 0)

(0, 0)

(0, 2)

(0, �4)

(0, 6)

(0, 3)

(0, �5)

1

��98

�

��12

�

�153�

undefined

undefined

0

0

1��23

��38

�

��25

��170��

47

�

�2

��23

�

��85

�

�190�

mathmatters3.com/self_check_quiz Lesson 13-1 The Standard Equation of a Circle 565

http://mathmatters3.com/self_check_quiz

Work with a partner to answer the following questions.

1. Study the graph shown at the right. From sixrandomly chosen points on the parabola, segmentshave been drawn to the parabola’s focus, point F.From these same points perpendicular lines havebeen drawn to line �, the directrix of the parabola.

2. Compare the lengths of the two segments from each point.

3. Describe the relationship between the points on the parabola and their distances from the focus andthe directrix.

BUILD UNDERSTANDING

A parabola is a set of points equidistant from a fixed point called the focus and a fixed line called the directrix.

Simple equations may be derived for parabolas that have a vertex at the origin and a directrix parallel to either the x-axis or the y-axis.

Let point P(x, y) be any point on the parabola such that FP � PD. Then use the distance formula.

FP � PD

�(x � 0�)2 � (y� � a)2� � �(x � x�)2 � (y� � a)2�

(x � 0)2 � (y � a)2 � (y � a)2 Square both sides. Simplify.

x 2 � y 2 � 2ay � a 2 � y 2 � 2ay � a 2

x 2 � 2ay � 2ay

x 2 � 4ay

When the focus (0, a) is on the y-axis and the directrix is y � �a, the equation for the parabola is x 2 � 4ay.

In the graph above, the variable a � 1. When a � 0, the parabola opens upward.When a � 0, the parabola opens downward.

GRAPHING Use a graphing calculator to graph x 2 � 4ay when a � 1 and a � �1. You will need to write the equation x 2 � 4ay in terms of y. Notice that they-axis remains the axis of symmetry whether the parabola opens upwards ordownwards.

13-2 More onParabolasGoals ■ Relate the equation of a parabola to its focus

and directrix.

Applications Satellite Communications, Energy

y

x

F

Directrix �

y

xF

D

P

(0, a)

(x, y)

y � �a

The distances are equal.


E x a m p l e 1

Find the focus and directrix of the parabolax 2 � �6y.

Solutionx 2 � �6y

x 2 � 4ay

4ay � �6y

4a � �6

a � ��32

�

x 2 � 4��32

��y

Because a is negative, the parabola opens downward. The focus is located at (0, a),

so the focus is �0, ��32

��. The directrix is

the line y � �a, so the directrix is

y � ��32

�� or y � �32

�.

E x a m p l e 2

SATELLITE COMMUNICATIONS A parabolic satellite dish directs all incoming signals to a receiver. The receiver is located at the vertex which is at the origin and the focus is at (0, 5). Find the simple equation for the parabola.

Solutionx 2 � 4ay

a � 5

x 2 � 4(5)y

x 2 � 20y

The equation is x 2 � 20y.

TRY THESE EXERCISES

Find the focus and directrix of each parabola.

1. x 2 � 12y 2. x 2 � �28y 3. 4x 2 � 32y � 0

4. GRAPHING Use a graphing calculator to graph the parabolas for Exercises1–3. Using the simple equation for a parabola x 2 � 4ay, where the focus is (0, a) and the directrix is y � �a, graph the directrix. How does the shape ofthe parabola change as the value of 4a increases?

Lesson 13-2 More on Parabolas 567

y

x

(0, a)

y � �a

y

x

(0, 5)

(0, 3), y � �3 (0, �7), y � 7(0, 2), y � �2

The parabola becomes flatter as the absolute value of 4a increases.mathmatters3.com/extra_examples


Find the equation for each parabola with vertex located at the origin.

5. Focus (0, 7) 6. Focus (0, �3) 7. Focus �0, �34

��8. WRITING MATH A parabola has its vertex at the origin. Explain how to use

the equation of the parabola to tell if the parabola opens up or down.

PRACTICE EXERCISES

Find the focus and directrix of each equation.

9. x 2 � 16y 10. x 2 � �10y

11. x 2 � 20y � 0 12. 2x 2 � �24y

13. 16y � 4x 2 � 0 14. �5x 2 � �30y

ENERGY Parabolic mirrors can be used to power steam turbines to generateelectricity. Three mirrors have the following focus points. Find the simpleequation for each. In each case, the vertex is located at the origin.

15. Focus (0, �2) 16. Focus (0, 9) 17. Focus �0, ��12��

Write the equation for each parabola with vertex at the origin and focus anddirectrix as shown.

18. 19.

20. 21.

22. CHAPTER INVESTIGATION Research the perihelion (closest orbital point to the sun) and aphelion (far-thest orbital point) for each planet of the Solar System. Make a rough sketch of the planets’ orbits and label the perihelion and aphelion for each planet.

y

x

x � �a

(a, 0)

y

x

x � a

(�a, 0)

y

x

(0, a)

y � �a

y

x

y � a

(0, �a)


x2 � 28y x2 � �12y

(0, 4), y � �4

(0, 5), y � �5

(0, 1), y � �1

x2 � �8y x2 � 36y x2 � �2y

x2 � �4ay x2 � 4ay

y2 � 4axy2 � �4ax

�0, ��52

��, y � �52

�

(0, �3), y � 3

�0, �32

��, y � ��

23�

x2 � 3y

Write the equation in the form x2 � 4ay and solve for a. If a � 0, theopen end is up. If a � 0, the open end is down.




The standard form for the equation of a parabolawith vertex (h, k) and axis parallel to the y-axis is(x � h)2 � 4a(y � k).

The standard form for the equation of a parabolawith vertex (h, k) and axis parallel to the x-axis is(y � k)2 � 4a(x � h).

Find the equation of each parabola.

23. Focus (2, 4) 24. Focus (4, 1) 25. Focus (2, 7)Vertex (2, 2) Vertex (2, 1) Directrix y � �3

26. WRITING MATH If a light is placed at the focus of a parabola, the rays willbe reflected off the parabolic surface parallel to the axis as shown. How isthis concept used in a flashlight?


Use x � ��2ba� to find the vertex. (Lesson 12-2)

27. y � x 2 � 3x � 6 28. y � 2x 2 � 5x � 8 29. y � x 2 � 3x � 4

30. �8 � y � 2x 2 � 2x 31. y � x 2 � 9 32. �x 2 � 3x � y � 5

33. y � 12 � x � 2x 2 34. 3x 2 � y � 6x � 9 35. y � x 2 � 4x � 3

36. 2x � y � �x 2 � 3 37. 4x 2 � �y � 3x � 8 38. 3x � �y � x 2 � 2

Let U � {1, 2, 3, 4, 5, 6, 7, 8, 9}, P � {1, 3, 4, 6, 7} and Q � {2, 5, 6, 7, 8}. Find eachunion or intersection. (Lesson 1-3)

39. P� 40. Q� 41. P � Q 42. Q � P

43. (P � Q)� 44. P� � Q� 45. Q� � P 46. Q� � P

47. (P � Q)� � (P � Q) 48. (P � Q) � (P� � Q�) 49. P� � (P � Q)

569

y

x

(h, k)

y

x

(h, k)

F

Lesson 13-2 More on Parabolas

(x � 2)2 � 8(y � 2) (y � 1)2 � 8(x � 2)

(1.5, 3.75)

(�0.5, 7.5)

(�0.25, 11.875)

(�1, �4) (�0.375, 8.5625)

(1.25, 4.875)

(0, �9)

(�1, �12)

(1.5, �0.25)

(�1.5, �6.25)

(�1.5, 2.75)

(�2, �7)

{2, 5, 8, 9} {1, 3, 4, 9} {6, 7}{1, 2, 3, 4, 5, 6, 7, 8}

{1, 3, 4}

{2, 5, 8}{1, 3, 4, 6, 7, 9}

{1, 2, 3, 4, 5, 8, 9}U

{9} {9}

(x � 2)2 � 20(y � 2)

Answers will vary. Sample: The smalllight bulb is surrounded by a mirroredparabolic surface that reflects light.

mathmatters3.com/self_check_quiz


PRACTICE LESSON 13-11. In the standard equation for a circle, what do the variables h and k represent?

2. What is the radius of a circle?

3. When is the standard equation of a circle x 2 � y 2 � r 2, r � 0?


4. radius 5 5. radius 3 6.center (0, 0) center (2, 4)

7. radius 4 8. radius 6center (�1, 2) center (�2, �3)


9. x 2 � y 2 � 13

10. (x � 2)2 � (y � 1)2 � 49

11. (x � 3)2 � (y � 4)2 � 25

12. x 2 � (y � 2)2 � 64

13. (x � 3)2 � y 2 � 36

14. (x � 4)2 � (y � 3)2 � 74

15. (x � 1)2 � y 2 � 28

16. (x � 3)2 � (y � 4)2 � 32

PRACTICE LESSON 13-2Determine if each statement is true or false.

17. When the focus is on the y-axis, and the directrix is y � a, the standardequation for the parabola is x 2 � 4ay.

18. The focus is equidistant from most points on the parabola.

19. When the y-coordinate of a focus is negative, the parabola opens downward.

Find the focus and directrix of each equation.

20. x 2 � �4y 21. x 2 � 13y 22. x 2 � 7y

23. x 2 � 3y � 0 24. x 2 � 5y � 0 25. 3x 2 � 27y � 0

26. x 2 � �8y 27. x 2 � 5y 28. x 2 � �3y

29. x 2 � 4y � 0 30. x 2 � 7y � 0 31. 2x 2 � 12y � 0

Find the standard equation for each parabola with vertex located at the origin.

32. Focus (0, 5) 33. Focus (0, �3) 34. Focus (0, 2)

35. Focus (0, �7) 36. Focus �0, �12

�� 37. Focus (0, 2.5)

y

x�5 5

5

�5


h is the x-coordinate of the center, and k is the y-coordinate of the center.

The radius is the distance from the center to any point on the circle.

when h � 0 and k � 0, that is, when the center is at the origin

(x � 2)2 � (y � 4)2 � 9

(x � 2)2 � (y � 3)2 � 36(x � 1)2 � (y � 2)2 � 16

r � �13�, c � (0, 0)

r � 5, c � (�3, 4)

r � 6, c � (3, 0)

r � 2�7�, c � (�1, 0)

r � 4�2�, c � (�3, 4)

focus � �0, �143��;

directrix is y � ��143�

focus � �0, �54

��;directrix is y � ��

54

�

27. focus � �0, �54


54

�

focus � �0, ��34

��;directrix is y � �

34

�

focus � (0, �2); directrix is y � 2

focus � (0, 1); directrix is y � �1 focus � �0, ��74

��; directrix is y � �74

� focus � �0, ��32

��; directrix is y � �32

�

focus � (0, �1);directrix is y � 1

focus � �0, �74


74

�

focus � �0, ��94


94

�

28. focus � �0, ��34


34

�

x2 � 20y

x2 � �28y

x2 � �12y

x2 � 2y

x2 � 8y

x2 � 10y

r � 7, c � (2, �1)

r � 8, c � (0, �2)

r � �74�, c � (4, �3)

false

false

false

x2 � y2 � 25

(x � 3)2 � y2 � 9

Review and Practice Your Skills

PRACTICE LESSON 13-1–LESSON 13-2Without graphing, determine whether each equation is that of a circle or aparabola. Assume r � 0. (Lessons 13-1–13-2)

38. x 2 � y 2 � r 2 39. x 2 � 4y 40. x 2 � r 2 � y 2

41. 3x 2 � 9y 42. 4x 2 � �8y 43. x 2 � r 2 � (y � 3)2

Find the radius and center for each circle. (Lesson 13-1)

44. x 2 � y 2 � 81 45. (x � 3)2 � (y � 4)2 � 36 46. (x � 2)2 � y 2 � 52

Write an equation for each circle. (Lesson 13-1)

47. radius 7 48. radius 5 49. radius 6center (0, �3) center (4, 2) center (�3, �1)

Chapter 13 Review and Practice Your Skills 571

circle

parabola

parabola

parabolacircle

circle

r � 2�13�, c � (2, 0)r � 6, c � (�3, 4)r � 9; c � (0, 0)

x2 � (y � 3)2 � 49 (x � 4)2 � (y � 2)2 � 25 (x � 3)2 � (y � 1)2 � 36

Workplace Knowhow

Career – Payload Specialist

mathmatters3.com/mathworks

Today astronauts are civilian and military specialists in scientific fields such as engineering. One particular type of astronaut is a payload specialist. A

payload specialist is a professional in the physical or life sciences and is skilled in working with equipment developed specifically for the space shuttle. Thepayload specialist also oversees experiments.

1. In order to perform the experiment, the space shuttle must be kept ahead of the moon’s orbit so that the ship, the Earth, and the moon form a right angle. The shuttle is orbiting the Earth at a distance of 325 km. The moon orbits the Earth at a distance of 384,403 km at its furthest distance (the current distance). Draw a diagram of the Earth, the moon and the ship forming a right angle. The right angle will fall at the center of the Earth. The Earth’s diameter is 12,756 km. The moon’s diameter is 3476 km. (Make your diagram look as if the moon is quite a bit farther away from the Earth than your ship.

2. How far is your ship from the center of the Earth? How far is the center of theEarth from the center of the moon?

3. Now that you have those two distances, draw the hypotenuse of the trianglebetween the Earth, the moon, and your ship. How far are you from the moon?

Payload specialists on space shuttle Discovery

Check students’ drawings.

390,838.229 km

6,703 km

392,519 km

http://mathmatters3.com/mathworks

Think of a line that intersects the coordinate plane at the origin.The right circular cones formed by rotating the line about the y-axisare used to study conics.

A conic section is formed by a plane intersecting the right circular cones.

P r o b l e m

Draw and name the conic section formed by each plane.

a. The plane is parallel to a side b. The plane is parallel to the of the cone and does not b. y-axis and does not passpass through the vertex. b. through the vertex.

Solve the Problem

a. b.

The conic section is a hyperbola.


13-3 Problem Solving Skills:Visual Thinking

y

xO

y

xO

The conic section is aparabola.

Problem SolvingStrategies

Guess and check

Look for a pattern

Solve a simplerproblem

Make a table, chartor list

Use a picture,diagram or model

Act it out

Work backwards

Eliminate possibilities

Use an equation orformula

✔

Math: Who,Where, When

Greek mathematicianand geometerApollonius of Pergastudied and named thecuts made by a flatplane as it intersects acone. About 225 B.C.,he wrote Conics, inwhich he described theproperties ofparabolas, circles,ellipses, andhyperbolas. Conicsections were laterfound to represent thepaths followed byplanets and projectiles.

parabola circle ellipse hyperbola

TRY THESE EXERCISESName the conic section formed by each plane.

1. The plane is parallel to the x-axis. 2. The plane does not contain a It does not contain the vertex. vertex or base. It is not parallel

to the x-axis.

Name the conic section or figure formed by each plane.

3. The plane intersects 4. The plane is perpendicular 5. The plane is parallel to aonly the vertex. to the x-axis and passes side of the cone and does not

through the vertex. pass through the vertex.

6. WRITING MATH Is it possible to intersect a double cone in such a way thattwo circles are formed? Explain.


Use a graphing calculator to determine the number of solutions for eachequation. For equations with one or two solutions, find the exact solutions byfactoring. (Lesson 12-3)

7. 0 � x 2 � 64 8. 0 � x 2 � x � 12 9. 0 � x 2 � 2x � 1

10. 0 � x 2 � 2x � 8 11. 0 � x 2 � 49 12. � x 2 � 5x � 6 � 0

13. x 2 � 5x � 6 14. 0 � x 2 � 25 15. x 2 � � 5x � 6

16. DATA FILE Use the data on measuring earthquakes on page 647. An earthquakein southern California measured 6.0 on the Richter scale. Another earthquake incentral California measured 3.0 on the Richter scale. How much more groundmovement occured in southern California than in central California? How muchmore energy was released? (Lesson 1-8)

y

xO

y

xO

y

xO

y

xO

y

xO

Lesson 13-3 Problem Solving Skills: Visual Thinking 573

Five-stepPlan

1 Read2 Plan3 Solve4 Answer5 Check

circle ellipse

pointtwo intersecting lines. parabola

See additional answers.

2; 8 and �8

2; �4 and 2

2; 6 and �1

2; �4 and 3

no solution

2; 5 and �5

1; �1

2; 6 and �1

2; �2 and �3

1000; 27,000

PRACTICE EXERCISES

Work with a partner.

You will need a piece of cardboard, two thumbtacks, string, andscissors. Place two thumbtacks in a piece of cardboard. Labeltheir positions F1 and F2. Let the distance between the two pointsbe 2c.

Tie the ends of a piece of string together to make a loop. Let thelength of the string be 2a � 2c where a is any quantity greaterthan c. Place the loop over the tacks. With a pencil held upright,keep the string taut and draw the ellipse.

Use the same loop of string and increase the distance between F1

and F2. Draw another ellipse.

1. How does the shape of the ellipse change as F1 and F2 arefarther apart?

2. For any point P on the ellipse, what is the sum of F1P � F2P?

3. If F1 and F2 were the same, what figure would you draw?

BUILD UNDERSTANDING

As you discovered in the activity above, an ellipse is defined by a point movingabout two fixed points. The two fixed points are called foci, the plural of focus.The sum of the distances from the two fixed points to any point on the ellipse is aconstant, F1P � F2P � 2a. If F1 and F2 are on the x-axis, the major axis ishorizontal. The major axis is vertical when F1 and F2 are on the y-axis.

An equation for the standard form of an ellipse can be derivedby placing the ellipse on a coordinate grid. Locate (0, 0) at themidpoint between F1 and F2. The distance from the center toeach focus is c. When x � 0, F1P1 � F2P1 and b 2 � a 2 � c 2.

Use the distance formula to determine the standard form for theequation of an ellipse. By letting y � 0, x-intercepts are (a, 0)and (�a, 0). By letting x � 0, y-intercepts are (0, b) and (0, �b).


13-4 Ellipses andHyperbolasGoals ■ Graph equations of ellipses and hyperbolas.

Applications Astronomy, Oceanography, Communications

F1 F22c

P

F1 F22c

y

xF1(�c, 0) F2(c, 0)

P (x, y)

P1(0, y)

(0, 0)c c

ba a

CheckUnderstanding

Why must length a begreater than length c?

It becomes flatter.

a circleIf a � c, there would notbe any slack. You couldnot draw the ellipse.

2a

Standard Form of Equation �ax2

2� � �by2

2� � 1 �ay2

2� � �bx2

2� � 1

Direction of Major Axis horizontal verticalFoci (c, 0), (�c, 0) (0, c), (0, �c)

Equations of Ellipses with Centers at the Origin

E x a m p l e 1

Graph the equation 4x 2 � 9y 2 � 36.

SolutionDivide both sides of the equation by 36 to change it to standard form.

4x 2 � 9y 2 � 36

� � 1

a 2 � 9, a � �3

b 2 � 4, b � �2

The x-intercepts are (3, 0) and (�3, 0). The y-intercepts are (0, 2)and (0, �2).

Locate the points and draw a smooth curve.

E x a m p l e 2

ASTRONOMY Jae is using a computer to model the orbit of amoon. After placing a grid over a drawing of the orbit, he finds thatthe foci of the ellipse are (4, 0) and (�4, 0), and the x-intercepts are(�5, 0) and (5, 0). He needs to enter the equation of the ellipse intothe computer to finish his work. Find the equation of the ellipse.

Solution

� � 1

a � �5, c � �4

b 2 � a 2 � c 2 Use the Pythagorean Theorem to find b2.

b 2 � 25 � 16

b 2 � 9

� � 1 Substitute in the standard form.

9x 2 � 25y 2 � 225 Multiply by 225 (9 � 25).

The equation of the ellipse with foci (4, 0) and (�4, 0) and x-intercepts (�5, 0)and (5, 0) is 9x 2 � 25y 2 � 225.

y 2

�9

x 2

�25

y 2

�b 2

x 2

�a 2

y 2

�4

x 2

�9

Lesson 13-4 Ellipses and Hyperbolas 575

Technology Note

To graph an ellipseusing a graphingcalculator, rewrite theequation in terms of y.

4x2 � 9y2 � 36 becomes

y � � �� 49

�x2� � 4�.

1. Enter the positive formof the equation as Y1to graph the upperpart of the ellipse.

2. To graph the lowerportion, let Y2 � �Y1.

3. Graph the ellipse. Youcan use the tracefeature to locate the x- and y-intercepts.

3

1

�1

�3

2 4�2�4

y

x

CheckUnderstanding

Write ellipse orhyperbola for eachequation.

1. �1x0

2

� � �y4

2

� � 1

2. �1x2

2

� � �y5

2

� � 1

3. 4x2 � 7y2 � 56

4. 8x2 � 13y2 � 104

1 and 4: ellipse;2 and 3: hyperbola





A hyperbola has some similarities to an ellipse. The distance from the center to avertex is a units. The distance from the center to a focus is c units. There are twoaxes of symmetry. The transverse axis is a segment of length 2a whose endpointsare the vertices of the hyperbola. The conjugate axis is a segment of length 2bunits that is perpendicular to the transverse axis at the center. The values of a, b,and c are related differently for a hyperbola than for an ellipse. For a hyperbola,c2 � a2 � b2.

E x a m p l e 3

ASTRONOMY Comets that pass by Earth only once may follow hyperbolic paths. Suppose a comet follows one branchof a hyperbola with center (0, 0) and foci on the y-axis if a � �15 and b � �20. Find the equation of the hyperbola.

Solution

Write the standard form of the equation of the hyperbola.

�ay2

2� � �bx2

2� � 1

�1y5

2

2� � �2x0

2

2� � 1 Substitute in the standard equation.

�22

y2

5� � �

4x0

2

0� � 1

400y 2 � 225x 2 � 90,000. Multiply by 90,000 (400 � 225).

The equation is 400y 2 � 225x 2 � 90,000.


Standard Form of Equation �ax2

2� � �by2

2� � 1 �ay2

2� � �bx2

2� � 1

Direction of Tranverse Axis horizontal verticalFoci (c, 0), (�c, 0) (0, c), (0, �c)Vertices (a, 0), (�a, 0) (0, a), (0, �a)Length of Transverse Axis 2a units 2a unitsLength of Conjugate Axis 2b units 2b units

Equations of Asymptotes y � ��ba

�x y � ��ba

�x

Equations of Hyperbolas with Centers at the Origin

y

xO

asymptote

conjugate axis

transverse axis

center

focus focus

asymptote

vertex vertex F1 F2

b c

a

The point oneach branchnearest thecenter is avertex.

As a hyperbolarecedes from itscenter, the branchesapproach linescalled asymptotes.

TRY THESE EXERCISES

Graph each equation.

1. 4x 2 � 16y 2 � 64 2. 4x 2 � 25y 2 � 100

3. Find the equation of the ellipse with foci (12, 0) and (�12, 0) and x-intercepts(13, 0) and (�13, 0).

4. Find the equation of the hyperbola with center (0, 0) and foci on the x-axis ifa � �9 and b � �6.

5. WRITING MATH Is a circle a type of ellipse? Explain your thinking.

PRACTICE EXERCISES


6. 9x 2 � 36y 2 � 36 7. 9x 2 � 16y 2 � 144

8. COMMUNICATIONS A communications satellite is launched into anelliptical orbit with foci (8, 0) and (�8, 0) and x-intercepts (10, 0) and (�10, 0). Find the equation of the ellipse.

9. OCEANOGRAPHY An object propelled through water travels along onebranch of a hyperbola with an experimental submarine at its center (0, 0)and in which a � 7 and b � 6 and the foci are on the x-axis. Find theequation for the hyperbola.

Graph each hyperbola.

10. 18y 2 � 8x 2 � 72 11. y 2 � x 2 � 144

Graph each ellipse.

12. 4x 2 � y 2 � 4 13. 25x 2 � 4y 2 � 100 14. 2x 2 � y 2 � 8


The ellipse shown has center (h, k) and axes parallel to the coordinate axes.

15. Substitute in the standard form for the equation of an ellipse with itscenter at the origin and foci on the x-axis to find an equation for anellipse with center (h, k).

16. Use your equation to find the center of the ellipse

16(x � 2)2 � 9(y � 1)2 � 144.


Solve by completing the square. (Lesson 12-4)

17. x 2 � 10x � 0 18. x 2 � 3x � 0 19. x 2 � 8x � 020. x 2 � 4x � 7 � 0 21. x 2 � 6x � 2 � 0 22. x 2 � 12x � 3 � 0

Lesson 13-4 Ellipses and Hyperbolas 577

y

x

(h, k)a

b


25x2 � 169y2 � 4225

36x2 � 81y2 � 2916


9x2 � 25y2 � 900


� � 1(y � k)2

�b2

(x � h)2

�a2

(�2, 1)

x � 0, x � �10

x � �2 ��11�

x � 0, x � �3

x � 3 ��7�

x � 0, x � 8

x � �6 ��39�

No, a circle is not an ellipse. The foci of an ellipse are different points.




36x2 � 49y2 � 1764


PRACTICE LESSON 13-3Draw and name the conic section formed by each plane.

1. The plane is parallel to a side of the 2. The plane intersects the vertex cone and does not pass through the b. and is parallel to the y-axis.vertex.

3. The plane is parallel to the x-axis and 4. The plane does not contain a vertex ordoes not contain the vertex. b. base, and is not parallel to the x-axis.

PRACTICE LESSON 13-45. Demonstrate how to determine the x-intercepts of an ellipse using the

standard form of the equation for an ellipse and letting y � 0.

6. Demonstrate how to determine the y-intercepts of an ellipse using thestandard form of the equation for an ellipse and letting x � 0.

7. The standard form of the equation of an ellipse and for a hyperbola aresimilar. Explain how they are different.


8. 4x 2 � 16y 2 � 64 9. 4x 2 � 16y 2 � 64 10. 9x 2 � 9y 2 � 81

11. 9x 2 � 9y 2 � 81 12. 25x 2 � 4y 2 � 100 13. 25x 2 � 4y 2 � 100

Find the equation of each ellipse.

14. foci (5, 0), (�5, 0) 15. foci (3, 0) and (�3, 0) 16. foci (4, 0) and (�4, 0)

x-intercepts (8, 0), (�8, 0) x-intercepts (10, 0), (�10, 0) x-intercepts (6, 0), (�6, 0)

Find the equation of each hyperbola. Assume all foci are on the x-axis.

17. center (0, 0) 18. center (0, 0) 19. center (0, 0)

a � 5, b � 3 a � 9, b � 4 a � 7, b � 4

y

xO

y

xO

y

xO

y

xO


parabola two triangles

ellipsecircle




39x2 � 64y2 � 2496

9x2 � 25y2 � 225 16x2 � 81y2 � 1296 16x2 � 49y2 � 784

91x2 � 100y2 � 9100 20x2 � 36y2 � 720

The standard equation for an ellipse uses the sum of thedistances from a point on the ellipse to the foci; the equationfor a hyperbola uses the differences of these distances.


PRACTICE LESSON 13-1–LESSON 13-4Name the conic section formed by the plane.(Lesson 13-3)

20. The plane is parallel to a side of the cone and does not passthrough the vertex.

Graph each equation. (Lesson 13-4)

21. 4x 2 � 4y 2 � 16 22. 4x 2 � 4y 2 � 16

23. 9x 2 � 36y 2 � 324 24. 9x 2 � 36y 2 � 324

25. 49x 2 � 16y 2 � 784 26. 49x 2 � 16y 2 � 784

Mid-Chapter QuizFind the radius and center for each circle. (Lesson 13-1)

1. (x � 3)2 � (y � 1)2 � 64 2. x 2 � (y � 6)2 � 6

3. (x � 7)2 � (y � 9)2 � 20 4. (x � 2)2 � y 2 � 18

Find the standard equation for each parabola with the vertex located at theorigin. (Lesson 13-2)

5. focus (0, 2) 6. focus (0, �6) 7. focus �0, �23

��8. focus �0, ��

14

�� 9. focus (6, 0) 10. focus (0.5, 0)

Write the standard equation for each ellipse. (Lesson 13-4)

11. foci (�2, 0) and (2, 0); x-intercepts (�5, 0) and (5, 0)

12. foci (�3, 0) and (3, 0); x-intercepts (�6, 0) and (6, 0)

Write the standard equation for each hyperbola. (Lesson 13-4)

13. center (0, 0); foci on x-axis; a � 4, b � 7

14. center (0, 0); foci on x-axis; a � 6, b � 8

Name the figure or conic section formed by the plane. (Lesson 13-3)

15. The plane intersects the side 16. The plane does not containof the cone and passes a vertex or base, and is through the vertex. not parallel to the x-axis.


y

xO

y

xO

y

xO

parabola

8, (3, �1) �6�, (0, �6)

2�5�, (�7, 9) 3�2�, (�2, 0)

x2 � 8y

x2 � �y

� � 1y2

�21

x2

�25

� � 1y2

�27

x2

�36

� � 1y2

�49

x2

�16

� � 1y2

�64

x2

�36

line circle

x2 � �24y

y2 � 24xy2 � 2x


x2 � �83

�y

Work with a partner to answer the following questions:

The table below shows the diameter and approximate circumference of some circles.

a. Find the ratio �Cd

� for each pair of values.

b. When d doubles, what happens to C?

c. Write an equation for C as a function of d.

d. Graph the function. Describe the graph and find its slope.

BUILD UNDERSTANDING

The circumference of a circle is a function of its diameter. This function can bewritten C � 3.14d. From the graph, you can see that the relationship betweendiameter and circumference is linear.

The relationship between diameter and circumference of a circle is an exampleof direct variation. The value of one variable increases as the other variableincreases.

Direct variation can be represented by an equation in the form y � kx, where k is a nonzero constant and x � 0. The constant k iscalled the constant of variation. For the example above,circumference varies directly with diameter. The constant ofvariation is 3.14.

If y varies directly as x, the constant of variation can be found if onepair of values is known.

E x a m p l e 1

What is the equation for a direct variation when one pair of values is x � 20 and y � 9?

Solutiony � kx Substitute in the equation for direct variation.

9 � k(20)

�290� � k

0.45 � k Solve for k.

The equation is y � 0.45x.


13-5 Direct VariationGoals ■ Solve problems involving direct variation.

Applications Food Prices, Space Exploration, Physics

Diameter (d) 3 6 9 12 24Circumference (C) 9.42 18.84 28.26 37.68 75.36

Reading Math

y � kx is read “y isdirectly proportional tox” or “y varies directlyas x.”

�Cd

� � 3.14

It doubles.

C � 3.14d

The graph is a straight line with slope 3.14.

Many examples of the use of direct variation may be found in everydaysituations.

E x a m p l e 2

FOOD PRICES The cost of apples varies directly withweight. If 9 lb of apples cost $4.32, how much will 17 lb ofapples cost?

Solutiony � kx

4.32 � k(9) Substitute.

�4.

932� � k

0.48 � k Solve for k.

y � 0.48x Write the equation.

y � 0.48(17) Substitute 17 for x.

y � 8.16 Solve.

Seventeen pounds of apples will cost $8.16

The equation for the area of a circle is A � �r2. Thearea varies directly as the square of the radius. This isan example of direct square variation. The equationrepresents a quadratic function.

Direct square variation may be expressed in the formy � kx 2 where k is a nonzero constant.

E x a m p l e 3

SPACE EXPLORATION An air filter used in a space vehicle is in the shape of acube. The surface area of a cube varies directly as the square of its sides. If thesurface area of an air filter with sides 12 in. long is 864 in.2, what is the surfacearea of an air filter in the shape of a cube with sides 11 in. long?

Solutiony � kx2

864 � k(12)2 Substitute in the equation.

864 � 144k Solve for k.

�81

64

44

� � k

6 � k

y � 6x 2 Write the equation.

y � 6(11)2 Substitute 11 for x.

y � 726 Solve.

The surface area of a cube with sides 11 in. in length is 726 in.2.

Lesson 13-5 Direct Variation 581

250

200

150

100

50

01 3 5 7 9

Radius

Are

a

CheckUnderstanding

k is sometimes calledthe constant ofproportionality. Howcould this problem besolved as a proportion?

�$4

9.32� � �

7x

�





TRY THESE EXERCISES

1. What is the equation of direct variation when one pair of values is x � 72 andy � 18?

2. If y varies directly as x and y � 60 when x � 50, find y when x � 15.

3. If y varies directly as x2 and y � 320 when x � 8, find y when x � 5.

4. POSTAGE Fifteen stamps cost $5.55. How much will 26 stamps cost?

5. EARNINGS A person’s income varies directly with the number of hours theperson works. If the pay for 16 h is $200, what is the pay for working 40 h?

6. CATERING Three vegetable platters serve a party of 20 people. How manyvegetable platters are needed for a party of 240 people?

PRACTICE EXERCISES

7. The distance (d) a vehicle travels at a givenspeed is directly proportional to the time (t) ittravels. If a vehicle travels 30 mi in 45 min,how far can it travel in 2 h?

8. ASTRONOMY The speed of a comet atperihelion (the closest orbital point to thesun) is 98,000 mi/h. How far will the comettravel in 30 sec?

9. BIOLOGY The expected increase (I) of apopulation of organisms is directlyproportional to the current population (n). Ifa sample of 360 organisms increases by 18, byhow many will a population of 9000 increase?

10. PHYSICS The distance (d) an object falls isdirectly proportional to the square of thetime (t) it falls. If an object falls 256 ft in 4 sec, how far will it fall in 7 sec?

11. SALES The cost of ribbon is directlyproportional to the length purchased. If9.5 yd of ribbon cost $3.42, how much will 14.75 yd cost?

12. SALES A person is paid $153 for 9 baskets of dried flowers. To earn $450,how many baskets of dried flowers must the person produce?

13. BAKED GOODS A bakery earns $33.75 profit on 9 cakes. How many cakesmust be sold for the bakery to make a profit of $150?

14. WRITING MATH A clothing store charges $1.75 in sales tax on an item thatcosts $25. The same store charges $2.80 in sales tax on a $40 item. Is sales taxan example of direct variation? Justify your answer.

15. ERROR ALERT The distance a spring will stretch, S, varies directly with theweight, W, added to the spring. A spring stretches 1.5 in. when 12 lb areadded. Paige plans to add 2 more pounds to the spring. She concludes thatthe spring will stretch 1.5 � 2, or 3.5 in. when the weight is added. Is Paige’sthinking correct? Explain your reasoning.


y � 0.25x

y � 18

y � 125

$9.62

$500

36

80 mi

816.67 mi

450

784 ft

$5.31

27

40

No, Paige’s conclusion is wrong. Using the directionvariation form, y � kx, if 14 lb are added to the spring, it will stretch 1.75 in.

Yes. Sales tax varies directly with thepurchase price of an item. In the example stated, the store is charging 7% sales tax on each purchase.


y � 18

The area of each regular polygon varies directly as the square of its sides. Onepair of values is given for each. Find the area for each regular polygon. Then findthe area of each polygon if one side is 9 units.

16. Pentagon 17. Hexagon 18. Octagons � 3 s � 5 s � 7A � 15.48 A � 64.95 A � 236.59

ELECTRICITY The number of kilowatts of electricity used by an appliance variesdirectly as the time the appliance is used.

19. If you watch television for 4.5 h, about how many kilowatt hours of electricity do you use?

20. If you dry your hair for 10 min, how many kilowatt hours of electricity do you use?

21. If the refrigerator runs for 2 h/day, how many kilowatt hours of electricity does it use?

22. If the water heater runs 2.75 h/day, how many kilowatt hours of electricity does it use?

23. If electricity costs 12.3¢ per kilowatt hour, find the cost for each activity in Exercises 19–22 to the nearest cent.

24. CHAPTER INVESTIGATION Choose a point within your school grounds orcommunity to represent the Sun. Using a map of your school or communityand an appropriate scale, plot the location of the aphelion and perihelion foreach planet on the map. Make a rough sketch of the orbits of the planets.


Write direct variation, direct square variation, or neither to describe how P variesas V increases or decreases in each equation.

25. P � 3V 26. P � �KV

�

27. MP � 2V 2 28. �18

�V 2 � P

29. �VP

� � 1 30. (4V � 1) � P � �1


Use the quadratic formula to solve each equation. (Lesson 12-5)

31. 2x 2 � 3x � 6 � 0 32. x 2 � 4x � 3 � 0 33. x 2 � 4x � 8 � 0

34. �2x 2 � 4x � 1 � 0 35. �4x 2 � 6x � 1 � 0 36. 2x 2 � 6 � 0

Find each value to the nearest hundredth. (Basic Math Skills)

37. �72� 38. �48� 39. �175� 40. �37�

Write each in simplest radical form. (Lesson 10-1)

41. (2�12�)(5�27�) 42. (2�8�)(3�12�) 43. 44. ��268��18�

��8�

Lesson 13-5 Direct Variation 583

A � 1.72s2, 139.32

1035

0.25

10

44

direct variation

direct square variation

direct variation

neither

direct square variation

neither

x � ��3 �

4�57��

x � ��2 �

2�6�

� x � ��3 �

4�13��

x � ��3�

x � 2 � 2�3�x � 3, 1

6.0813.236.938.49

180 24�6��32

� ��

342��

19. $0.13; 20. $0.03; 21. $1.23; 22. $5.41

A � 2.598s2, 210.438 A � 4.828s2, 391.068


Appliance Kilowatts per hourLight bulbs 0.001 per wattElectric blanket 0.07Stereo 0.1Color television 0.23Hair dryer 1.5Refrigerator 5.0Iron 1.0Freezer 3.0Water heater 16.0

Approximate Kilowatt Usage of Some Appliances



Work with a partner to answer the following questions.

The table shows the cost per person of renting a vacation home.

a. Find the product of nc for each pair of values.

b. When n doubles, what happens to c?

c. Write an equation for c as a function of n.

d. Graph the function. Explain how c varies as n increases.

BUILD UNDERSTANDING

The cost per person for renting the vacation home is afunction of the number of people. As the number of people increases, the cost per person decreases. Therelationship between n and c is an example of inversevariation. The value of one variable decreases as the valueof the other increases.

Inverse variation may be represented by an equation in the form y � �

kx

�, where k is a nonzero constant and x � 0.For the example above, the cost per person of renting thevacation home varies inversely as the number of peoplesharing the cost. The constant of variation is $800. Thegraph you drew in the activity above illustrates the graphof an inverse variation.

Most applications involve only the part of the graph thatlies in the first quadrant.

13-6 Inverse VariationGoals ■ Solve problems involving inverse variation and

inverse square variation.

Applications Astronomy, Physics, Travel


Number of people (n) 1 2 4 5 8 10Cost per person (c) $800 $400 $200 $160 $100 $80

$800

$700

$600

$500

$400

$300

$200

$100

1 2 3 4 5 6 7 8 9 10Number of people

Co

st p

er p

erso

n

Reading Math

y � �kx

� is read: “y is inversely proportionalto x” or “y variesinversely as x.”

$800

It is halved.

c � �$8

n00�

As n increases, c decreases.

E x a m p l e 1

Write an equation in which y varies inversely as x if one pair of values is y � 240and x � 0.4.

Solutiony � �

kx

� Substitute in the equation for inverse variation.

240 � �0k.4�

240 � 0.4 � k

96 � k Solve for k.

The equation is y � �9x6�.

Travel time varies inversely as travel speed. In other words, travel time decreasesas speed increases.

E x a m p l e 2

ASTRONOMY At its greatest distance from the sun, an asteroid travels a certaindistance in 40 min while traveling at 250 mi/h. How long would it take theasteroid to travel the same distance, traveling at 400 mi/h?

Solutiony � �

kx

� Find an equation.

40 � �2

k50�

40(250) � k

10,000 � k

y � �10,

x000�

y � �10

4,00000� Substitute 400 into the equation.

y � 25 Solve for y.

The trip will take 25 min travelling at 400 mi/h.

The table below shows how the brightness (in lumens) of a 60-watt light bulbvaries with distance from the bulb.

This is an example of inverse square variation. The brightness of the light variesinversely as the square of the distance from its source.

Inverse square variation can be expressed in the form

y � or x 2y � k, where k is a nonzero constant and x � 0.

For the example above, l � . The constant of variation is 880.880�d 2

k�x 2

Lesson 13-6 Inverse Variation 585

Distance (in feet) 1 2 4 8Brightness (in lumens) 880 220 55 13.75



E x a m p l e 3

PHYSICS The brightness of a light bulb varies inversely as the squareof the distance from the source. If a light bulb has a brightness of 400lumens at 2 ft, what will be its brightness at 20 ft?

Solutionx2y � k

(2)2(400) � k

4(400) � k

1600 � k Solve.

x2y � 1600 Write the equation.

(20)2(y) � 1600 Substitute 20 for x.

400y � 1600

y � 4 Solve.

At 20 feet, the brightness will be 4 lumens.

TRY THESE EXERCISES

1. Write an equation in which y varies inversely as x if one pair of values is y � 85 and x � 0.8.

2. If y varies inversely as x and one pair of values is y � 44 and x � 5, find y when x � 8.

3. In some cities, the amount paid by each person sharing a cab varies inverselyas the number of people who share the cab. If 2 people pay $4.50 each for aride, how much will the same ride cost 5 people?

4. If y varies inversely as the square of x and y � 224 when x � 2, find y when x � 8.

5. PHYSICS The brightness of a light bulb varies inversely as the square of thedistance from the source. If a light bulb has a brightness of 300 lumens at2 ft, what will be its brightness at 10 ft?

PRACTICE EXERCISES


7. If y varies inversely as x and one pair of values is y � 39 and x � 3, find ywhen x � 39.

8. If y varies inversely as x and one pair of values is y � 12 and x � 10, find ywhen x � 20.

9. WRITING MATH Think of a real-life example of inverse variation. Explainhow you know the type of variation the example represents.

10. TRAVEL If it takes 30 min to drive from Ann’s house to the museumtraveling at 40 mi/h, how long will it take traveling at 50 mi/h?


Substitute known values into theequation for inverse square variation.

y � �6x8�

y � 27.5

$1.80 each

y � 14

12 lumens

y � �22

x7.5�

y � 3

y � 6

Answers may vary. Studentsshould mention that in an inverse function, as one variable increases, the other decreases.

24 min.


y � 27.5



11. MAGNETISM The force of attraction between two magnets varies inverselyas the square of the distance between them. When two magnets are 2 cmapart, the force is 64 newtons. What will be the force when they are 8 cmapart?


Write an equation of inverse variation for each.

13. Barometric pressure (p) is inversely proportional to the altitude (a).

14. The time (t) required to fill a swimming pool is inversely proportional to thesquare of the diameter (d) of the hose used to fill it.

15. The current (I) flowing in an electric circuit varies inversely as the resistance(R) in the circuit.

16. The intensity of the heat from a fireplace varies inversely as the square of thedistance from the fireplace. Your friend is next to the fireplace. If you onlyfeel �

116� the amount of heat that your friend feels, how much farther are you

from the fireplace than your friend?


When a quantity varies directly as the product of two or more other quantities,the variation is called a joint variation. If y varies jointly as w and x, then y � kwx.

When a quantity varies directly as one quantity and inversely as another, thevariation is called a combined variation. If y varies directly as w and inversely as x, then y � �

kxw�.

Write an equation of joint variation for each.

17. m varies directly as s and t.

18. a varies jointly as c and d.

19. r varies jointly as w, x, and y.

Write an equation of combined variation for each.

20. n varies directly as t and inversely as e.

21. v varies directly as r and inversely as the square of w.

22. d varies directly as the square of a and inversely as b.


Calculate the distance between each pair of points. Round to the nearesthundredth if necessary. (Lesson 12-6)

23. A(3, 2), B(1, �8) 24. C(�6, 2), D(5, �9) 25. E(3, �4), F(8, �6)

26. G(�4, �3), H(8, �1) 27. J(7, �5), K(3, 8) 28. L(�4, �7), M(�2, �3)

In the figure, R is the midpoint of QS. Find each measure. (Lesson 3-3)

29. QR 30. RS 31. ST

4x � 1 6x � 4 7x � 3

Q R S T

Lesson 13-6 Inverse Variation 587

4 newtons

y � 64

p � �ak

�

t �k

�d 2

I � �Rk

�

m � kst

a � kcd

r � kwxy

n � �ket�

v � �wkr

2�

d �ka2

�b

10.20

12.17

15.56

13.60

5.39

4.47

20.51111

4 times farther away



PRACTICE LESSON 13-51. In a direct variation, if y decreases, what must be true about x?

2. In the equation representing direct variation, y � kx, what must be trueabout k?

3. What is the difference between a graph showing a direct variation and agraph showing a direct square variation?

Write the equation of direct variation using the given values.

4. x � 5, y � 20 5. x � 12, y � 8 6. x � 9, y � 10

7. x � 16, y � 4 8. x � 24, y � 4 9. x � 8, y � 18




13. If y varies directly as x 2, and y � 12 when x � 4, find y when x � 8.



16. A survey showed that 7 out of 10 people liked the taste of Shimmertoothpaste. At this rate, how many people out of 2400 would be expected to like the taste of Shimmer?

PRACTICE LESSON 13-617. Refer to page 584. The example of an inverse variation was given as the cost

of a vacation home per person. Why is the application only concerned withthe part of the graph that lies in the first quadrant?

18. Write an equation in which y varies inversely as x if one pair of values is y � 4and x � 12.

Find answers to the nearest hundredth.

19. Write an equation in which y varies inversely as x if one pair of values is y � 7and x � 0.9.

20. If y varies inversely as x and y � 4 when x � 2, find y when x � 7.

21. If y varies inversely as x and y � 7 when x � 4, find y when x � 9.


Write an equation of inverse variation for each.

23. The time (t) it takes to travel from one point to another is inverselyproportional to the speed (s) of the travel.

24. The amount of oxygen (o) in the air is inversely proportional to the altitude (a).

588 Chapter 13 Advanced Functions and Relations

k � 0

y � 4x

y � �14

�x

y � �23

�x

y � �16

�x

y � �190�x

y � �94

�x

y � 5

y � 2�17

�

y � 14

y � 48

y � 324

y � 405

1680

y � �4x8�

y � �6x.3�

y � 1.14

y � 3.11

y � 4

t � �ks

�

o � �ak

�

In real life, there are no negative people andno negative costs. Quadrant I is the only quadrant where both variables are positive.

x decreases if the constant of variation is positive.

A direct variation graph is linear; a direct square variation graph is a quadratic function.


PRACTICE LESSON 13-1–LESSON 13-6Graph each equation. (Lessons 13-1–13-2)

25. (x � 2)2 � (y � 3)2 � 49 26. x 2 � �5y 27. x 2 � 4y

28. (x � 2)2 � y 2 � 25 29. (x � 4)2 � (y � 3)2 � 64 30. x 2 � (y � 5)2 � 36

Write the equation of direct variation using the given values. (Lesson 13-5)

31. x � 10, y � 25 32. x � 9, y � 3 33. x � 5, y � 7

34. One survey showed that 7 out of 12 eligible people in Greenville planned tovote in the next election. If the population of eligible voters in Greenville is19,824, how many could be expected to vote?



y � �52

�x y � �13

�x y � �75

�x

11,564

Workplace Knowhow

Career – Astronomer

mathmatters3.com/mathworks

Astronomers work in a sub-field of physics. Astronomers study things such asthe birth of stars, the death of stars, natural satellites, the composition of

planets, and the possibility of life on other planets. Astronomers work inobservatories with very large telescopes, in universities teaching and inplanetariums. Astronomers also work with engineers in the design, launch, anduse of deep space probes and satellites that send astronomical data back to theEarth from planets too far away and too inhospitable for humans to visit. Precisecalculations for the orbits of the satellites and probes are essential for theirmissions’ success.

1. Suppose that you discovered a new solar system with an elliptical orbit ofplanets around two suns. What would be the mathematical term for thelocation of each sun?

The suns are observably 5 in. apart in scaled distance. You also observe the 4planets’ orbits for a period of six months for their measure. The point at whichthe planets stop moving away from the suns and start moving back toward themis the x-intercept of the ellipse when their orbits are graphed. These distancesfrom one of the suns are as follows:

a. Planet A is 1.2 in. from a sun b. Planet B is 2.4 in. from a sun

c. Planet C is 3.8 in. from a sun d. Planet D is 5.2 in. from a sun

2. For graphing an ellipse around the origin on the coordinate plane, determine x-intercepts of each planet.

3. Calculate the y-intercept of each orbit.

4. Write the equation for each planet’s elliptical orbit.

The locations of the suns are called foci.

A: � 3.7; B: � 4.9; C: � 6.3; D: � 7.7

A: 2.728; B: 4.214; C: 5.783; D: 7.283


http://mathmatters3.com/mathworks

Work with a partner. You may use a graphing calculator.

The graph of a quadratic function divides the coordinate plane into three sets ofpoints. Graph y � x 2 � 2x � 1 on a coordinate plane.

a. Find 5 ordered pairs for which y � x 2 � 2x � 1.

b. Find 5 ordered pairs for which y � x 2 � 2x � 1.

c. Find 5 ordered pairs for which y � x 2 � 2x � 1.

d. Use the points you found for b to help you locate and shade the region of thegraph where y � x 2 � 2x � 1.

e. Use the points you found for c to help you locate and draw horizontal linesthrough the region of the graph where y � x 2 � 2x � 1.

BUILD UNDERSTANDING

Just as you used linear equations to graph linear inequalities, you can usequadratic equations to graph quadratic inequalities.

E x a m p l e 1

Graph �y 2 � 4x 2 a 16.

SolutionGraph the hyperbola �y 2 � 4x 2 � 16. Use the intercepts of (2, 0),(�2, 0), (0, 4), and (0, �4) to draw the rectangle. Then draw theasymptotes and sketch the hyperbola.

Because �y 2 � 4x � 16 is not part of the solution, the hyperbolais drawn with a dashed line.

To decide which points are part of the solution set, select pointson the graph and substitute their coordinates into the equation.

Select (0, 0): �(0)2 � 4(0)2 � 16

0 � 0 � 16

Select (�3, 1): �(1)2 � 4(�3)2 � 16 Select (4, 0): �(0)2 � 4(4)2 � 16

�(1) � 4(9) � 16 0 � 4(16) � 16

35 � 16 64 � 16

These points and the regions they contain are in the solution set. The solution setis the shaded region shown on the graph.

590

13-7 Quadratic InequalitiesGoals ■ Graph quadratic inequalities.

Applications Astronomy, Communications, Computer Design


6

3

�6

y

x�3 3 6�6

The point (0, 0) and the region that contains it are not in the solution set.



Systems of inequalities can be solved by finding the intersections of their graphs.

E x a m p l e 2

ASTRONOMY Radio commands may be sent to a space probe during a specific portion of its flight. If commands are sent too soon or too late, they will not be received by the probe. The solution set of the following system of inequalities is used to determine when commands may be sent. Solve the system of inequalities by graphing.

9x 2 � 4y 2 � 36

y � x � 2

SolutionGraph the ellipse 9x 2 � 4y � 36. The center is at the origin. The x-intercepts are(2, 0) and (�2, 0). The y-intercepts are (0, 3) and (0, �3).

The points on the ellipse are not in the solution set, so theellipse is drawn with a dashed line. Point (0, 0) is in thesolution set, so points inside the ellipse are part of thesolution set.

Graph y � x � 2. The solution set includes the line, so theline is solid. For the region above the line, y � x � 2, so partof this region is in the solution set.

The intersection of the two equations is shown by the greenregion of the graph.

E x a m p l e 3

Solve this system of inequalities by graphing.

x 2 � y 2 X 49

y a 2x 2 � 2

SolutionGraph the circle with radius 7. The circle is in the solution set; draw the circle with a solid line.Point (0, 0) is in the solution set. The regioninside the circle is in the solution set.

Graph parabola y � 2x 2 � 2. The parabola is notin the solution set; graph the parabola with adashed line. Point (0, 0) is not in the solution set,so the region inside the parabola is in thesolution set.

The region shaded green shows the intersection of the two equations.

Lesson 13-7 Quadratic Inequalities 591

5

1

�1

�5

y

x1 3�1�3 5�5

8

�4

�8

y

x4�4�8 8

CheckUnderstanding

How can you check tobe sure that the regioninside the parabola isin the solution set?

Substitute pointsthat lie inside theparabola into theequation.

Interactive Labmathmatters3.com



TRY THESE EXERCISES

Graph each inequality.

1. x 2 � y 2 � 81 2. y � 2x 2 � 1

3. 3x 2 � 12y2 � 48 4. 9x 2 � 25y 2 � 225

Graph each system of inequalities.

5. (x � 4)2 � (y � 2)2 � 4 6. y � x 2 � 3x �4

y � x � 6 (x � 1)2 � (y � 3)2 � 4

7. 4x 2 � 25y 2 � 100 8. 25x 2 � 4y 2 � 100

x 2 � y 2 � 25 x � �1

PRACTICE EXERCISES


9. 4x 2 � 16y 2 � 64 10. (x � 5)2 � (y � 1)2 � 4

11. y � x 2 � x � 1 12. 9x 2 � 16y 2 � 144

Graph each system of inequalities.

13. x 2 � y 2 � 25 14. y � x 2 � 2x

x 2 � y 2 � 100 y � x � 3

15. (x � 1)2 � (y � 3)2 � 25 16. x 2 � 16y 2 � 16

y � x 2 � 2x � 1 x 2 � y 2 � 64

Use the inequalities you have graphed in this lesson to help you complete eachstatement. Use b, a, or �.

17. If x 2 � y 2 ___?__ r 2, points inside the circle are in the solution set.

18. If � ___?__ 1, points outside the ellipse are in the solution set.

19. If ax 2 � bx � c ___?__ y, the region inside the parabola is the solution set.

20. COMMUNICATIONS The limits of a transmitter can be modeledusing the system of inequalities below. Graph the system anddescribe the solution set.x 2 � y 2 � 36

� � 1

21. COMPUTER DESIGN The system of inequalities below defines thecapabilities of a computer chip. Graph the system and describe thesolution set.

y � x 2 � 2x � 3

y � x 2 � x

y 2

�36

x 2

�100

y 2

�b 2

x 2

�a 2

592 Chapter 13 Advanced Functions and Relations

Problem SolvingTip

All hyperbolas andellipses in these exerciseshave center (0, 0).

For 1–4, see additional answers, p. 591.




�

�

�

Circle with center (0, 0) and r 6; ellipse with center(0, 0) and x-intercepts 10, y-intercepts 6; solutionset is all points inside both figures.

The graph of the solution set shows points above thefirst parabola and below the second parabola.


22. CHAPTER INVESTIGATION Select landmarks at your school or within yourcommunity to represent aphelion, or farthest orbital point, for eachlandmark. Mark these landmarks on your map. Share your findings with theclass, and discuss how this activity has increased your understanding of thesize of the solar system.


The solution of the inequality x 2 � 8x � �12 is shown on the number lines below.

23. WRITING MATH Explain this method for solving quadratic inequalities.

24. Does the solution check? Try these points: 1, 3, 5, 8.

25. What is the solution to the given inequality?

26. Use this method to solve x 2 � 8x � �15.


Calculate the midpoint of the segment with the given endpoints. (Lesson 12-6)

27. A(�6, 3), B(4, �5) 28. C(3, �2), D(�8, 9) 29. E(1, 0), F(�3, �7)

30. G(3, 0), H(�8, 2) 31. J(2, 7), K(�7, 2) 32. L(4, �5), M(1, 3)

33. N(0, 7), P(3, �4) 34. Q(1, 6), R(�6, �1) 35. S(8, �3), T(5, �6)

36. U(�6, 4), V(�3, �5) 37. W(�2, 0), X(1, 6) 38. Y(3, 5), Z(8, 2)

Each figure below is a parallelogram. Find a and b. (Lesson 4-8)

39. 40. 41.

Find the volume of each figure to the nearest whole number. (Lesson 5-7)

42. 43. 44.

45. Solve by completing the square: x 2 � 20x � 1 � 0. (Lesson 12-4)

3.4 m

7.6 m3 ft

4 ft

2 ft

7 ft

5 ft

8 ft

1 ft8 cm

16 cm

a° b°

60°

a° b°

109°

a° b°

75°

Lesson 13-7 Quadratic Inequalities 593

x � 2 � 0

x � 6 � 0

(x � 2)(x � 6) � 0

0�1�2�3 21 3 4 5 6 7 8

0�1�2�3 21 3 4 5 6 7 8

0�1�2�3 21 3 4 5 6 7 8

yes

(�1, �1)

��32

�, �32

��

92

�, ��12

�� 12

�, 3�

��52

�, �72

��

52

�, �92

��

52

�, �52

��

��1, ��72

��

��123�, ��

92

��

121�, �

72

��

a � 105, b � 75 a � 71, b � 109

a � 120b � 60

2 � x � 6

3 � x � 5

x � �10 � �101�

276 m3305 ft31072 cm3


Answers will vary.

��52

�, 1� ��52

�, �1�




13-8 Exponential FunctionsGoals ■ Graph exponential functions.

■ Solve problems involving exponential growth and decay.

Applications Population, Investments, Transportation

Work with a partner. You will need graphing paper.

1. Copy and complete the following table.

2. Use the table to write seven ordered pairs of the form (exponent, value of expression). ��2, �

14

��, ��1, �12

��, (0, 1), (1, 2), (2, 4), (3, 8)

3. Locate the seven ordered pairs on a coordinate plane. Then draw a smoothcurve through the points. See additional answers.

4. Describe the graph. Where does the curve cross the y-axis? See additional answers.

BUILD UNDERSTANDING

The graph you drew in the activity above is the graph of y � 2x. This type offunction, in which the variable is the exponent, is called an exponential function.

E x a m p l e 1

Graph y � 2x � 3. State the y -intercept.

SolutionYou can use a graphing calculator to graph y � 2x � 3. The y-intercept is �2.

Expression Exponent Value of Expression

2�2 �2 �14�

2�1 �1 �12

�

20 0 121 1 222 2 423 3 8

E x a m p l e 2

Compare the graphs of y � 3x and y � ��13

��x.

SolutionMake a table for y � 3x. Graph the function.

Make a table for y � ��13

��x

. Graph the function.

The graphs are reflections of each other across the y-axis. Both graphs have a y-intercept of 1. The graph of y � 3x increases from left to right. The graph of

y � ��13

��xdecreases from left to right.

Two examples of exponential functions are exponential growth and exponentialdecay. Exponential growth happens when a quantity increases by a fixed rate eachtime period. Exponential decay happens when a quantity decreases by a fixed rateeach time period. The general equations for exponential growth and decay aregiven below.

Exponential growth y � C(1 � r)t

Exponential decay y � C(1 � r)t

In these equations, y represents the final amount. C represents the initialamount, r represents the fixed rate or percent of change expressed as a decimal,and t represents time.

E x a m p l e 3

POPULATION The country of Latvia has been experiencing a 0.6% annualdecrease in population. In 2003, its population was 2,350,000. What would youpredict the population to be in 2013 if the rate remained the same?

SolutionSince the population is decreasing by a fixed rate each year, this is an example ofexponential decay.

y � C(1 � r)t

y � 2,350,000(1 � 0.006)10 Substitute in the equation.

y � 2,212,747 Use a calculator to solve for y.

In 2013, the population will be about 2,212,750.

Lesson 13-8 Exponential Functions 595

x �3 �2 �1 0 1 2 3

y �217� �

19

� �13

� 1 3 9 27

x �3 �2 �1 0 1 2 3

y 27 9 3 1 �13

� �19

� �217�


�5

20

105

15

253035

�2 �1�4�3 21 3 4

y

x

y � 13( (x

y � 3x


TRY THESE EXERCISES

Graph each function. State the y-intercept. 1–3. See additional answers for graphs.

1. y � 5x 1 2. y � 2 � 3x 2 3. y � 2x � 4 5

4. INVESTMENTS A municipal bond pays 5% per year. If $2000 is invested inthese bonds, find the value of the investment after 4 yr. about $2431.01

5. FARMING A farmer buys a tractor for $60,000. If the tractor depreciates 10%per year, what is the value of the tractor after 8 yr? about $25,828


Graph each function. State the y-intercept. 6–14. See additional answers for graphs.

6. y � 4x 1 7. y � 10x 1

8. y � ��12

��x

1 9. y � ��110��

x

1

10. y � 5(2x ) 5 11. y � 3(5x ) 3

12. y � 3x � 7 �6 13. y � 3x � 6 7

14. y � 2(3x ) � 1 1

15. TECHNOLOGY Computer use around the world has risen 19% annuallysince 1980. If 18.9 million computers were in use in 1980, predict the numberof computers that will be in use in 2015. about 8329.2 million computers

16. POPULATION The population of Mexico has been increasing at an annualrate of 1.7%. If the population of Mexico was 104,900,000 in the year 2003,predict its population in 2015. about 128,418,302

17. VEHICLE OWNERSHIP A car sells for $24,000. If the annual rate ofdepreciation is 13%, what is the value of the car after 8 yr? about $7877

18. NUTRITION A cup of coffee contains 130 mg of caffeine. If caffeine iseliminated from the body at a rate of 11% per hour, how much caffeine willremain in the body after 3 h? about 92 mg

19. INVESTMENTS Determine the amount of an investment if $3000 is investedat an interest rate of 5.5% each year for 3 yr. about $3522.72

20. REAL ESTATE The Villa family bought a condominium for $115,000.Assuming that the value of the condo will appreciate 5% each year, howmuch will the condo be worth in 6 yr? about $154,111

21. BUSINESS A piece of office equipment valued at $35,000 depreciates at asteady rate of 10% per year. What is the value of the equipment in 10 yr?

22. VEHICLE OWNERSHIP Carlos needs to replace his car. If he leases a car, hewill have an option to buy the car after 2 yr for $14,458. The current price ofthe car is $17,369. If the car depreciates at 16% per year, how will thedepreciated price compare with the buyout price of the lease?


about $12,204

The buyout price will be greater than the depreciated price.

23. ERROR ALERT Amanda graphed y � ��14

��x

at the right. Is she correct? Explain.

CRITICAL THINKING For Exercises 24–26, describe the graph of each equation as a transformation of the graph of y � 4x.

24. y � ��14

��x

25. y � 4x � 2 26. y � 4x � 6


In general, for any real number b and for any positive integer n,

b � �nb�, except when b � 0 and n is even.

Write each expression in radical form.

27. 6 �5 6� 28. 26 �4 26� 29. m �6 m�

Write each radical using rational exponents.

30. �17� 17 31. �325� 25 32. �8

x� x

33. WRITING MATH Explain why (�9) is not a real number.


Factor each trinomial. (Lesson 11-7)

34. d2 � 13d � 12 (d � 1)(d � 12) 35. t2 � 8t � 12 (t � 2)(t � 6)

36. p2 � 2p � 24 (p � 4)(p � 6) 37. s2 � 8s � 20 (s � 2)(s � 10)

38. a2 � ab � 2b2 (a � b)(a � 2b) 39. m2 � 5mn � 6n2 (m � 2n)(m � 3n)

Simplify. (Lesson 11-2)

40. (x3y4)(xy3) x4y7 41. (�3mn2)(5m3n2) �15m4n4

42. 3b(5b � 8) 15b2 � 24b 43. �12

�x(8x � 6) 4x2 � 3x

44. 5y(y2 � 3y � 6) 5y3 � 15y2 � 30y 45. �ab(3b2 � 4ab � 6a2) �3ab3 � 4a2b2 � 6a3b

Use the Pythagorean Theorem to find the unknown length. Round youranswers to the nearest tenth. (Lesson 10-2)

46. 47. 48.

4 ft

8 ft

6 cm

2 cm

5 m

4 m

1�2

1�8

1�3

1�2

1�6

1�4

1�5

1�n

Lesson 13-8 Exponential Functions 597

reflection over the y-axis

In radical form, the expression would

be ��16�, which is not a real number because the radicand is negative.

translation 2 units up translation 6 units downreflection over the y-axis



2 4

2

4

�2�4

�2

�4

y

x

3 m6.3 cm

8.9 ft


PRACTICE LESSON 13-7Graph each inequality. 1–6. See additional answers.

1. 4x2 � 2y2 � 8 2. y � �18

�x2

3. x2 � 4y2 � 4 4. x2 � y2 � 25

5. (x � 2)2 � (y � 3)2 � 9 6. y � x2 � 4x

Graph each system of inequalities. 7–15. See additional answers.

7. x � 2y � 1 8. x2 � y2 � 4 9. x � y � 4

x2 � y2 � 25 4y2 � 9x2 � 36 9x2 � 4y2 � 36

10. x2 � y2 � 25 11. x � y � 2 12. x2 � y2 � 36

4x2 � 9y2 � 36 4x2 � y2 � 4 4x2 � 9y2 � 36

13. y2 � x 14. x2 � y 15. y � x

x2 � 4y2 � 16 y2 � x2 � 4 y � x2 � 4

PRACTICE LESSON 13-8Graph each function. State the y-intercept. 16–21. See additional answers for graphs.

16. y � ��14

��x

1 17. y � 9x 1 18. y � 0.5(4x) 0.5

19. y � 2��12

��x

2 20. y � ��13

��x

� 3 �2 21. y � 2x � 5 �4

22. VEHICLE OWNERSHIP A pick-up truck sells for $27,000. If the annual rate ofdepreciation is 12%, what is the value of the truck after 5 yr? about $14,249

23. INVESTMENTS Determine the amount of an investment if $5000 is investedat an interest rate of 4.5% each year for 4 yr. about $5962.59

24. POPULATION In 2003, the population of Jamaica was 2,696,000. If thepopulation increases at a rate of 1.2% per year, predict its population in 2015.

25. COMPUTER COSTS A computer package costs $1800. If it depreciates at arate of 18% per year, find the value of the computer package after 3 yr.

26. POPULATION The population of Ukraine has been decreasing by an annualrate of 0.9%. The population of Ukraine was 48,055,000 in 2003. Predict itspopulation in 2010. about 45,108,061

27. INCOME The median income was $32,000 in 2003. If income increases at arate of 0.5% per year, predict the median income in 2013. about $33,636

28. REAL ESTATE A house is purchased for $180,000. If the value of the houseincreases 4.5% per year, what is its value after 8 yr? about $255,978



about 3,110,900

about $992

PRACTICE LESSON 13-1–LESSON 13-8Write an equation for each circle. (Lesson 13-1)

29. radius 12 30. radius 6 31. radius 2center (�1, 3) center (5, 4) center (�4, �1)

Find the radius and center for each circle. (Lesson 13-1)

32. x2 � (y � 2)2 � 16 33. (x � 1)2 � (y � 5)2 � 1 34. (x � 7)2 � (y � 4)2 � 49

Find the focus and directrix of each equation. (Lesson 13-2)

35. x2 � 20y 36. 3x2 � �24y 37. 28y � 7x2 � 0

38. Name the conic section formed by the plane. The plane at the right is parallel to the x-axis and does not contain the vertex.(Lesson 13-3) circle

Graph each equation. (Lesson 13-4) 39–44. See additional answers.

39. x2 � 36y2 � 36 40. x2 � 2y2 � 2 41. x2 � y2 � 4

42. 4x2 � 8y2 � 32 43. 3x2 � 9y2 � 27 44. 27x2 � 9y2 � 81

45. If y varies directly as x and y � 15 when x � 3, find y when x � 12. (Lesson 13-5) 60



48. If y varies inversely as x and y � 5 when x � 10, find y when x � 2. (Lesson 13-6) 25

49. If y varies inversely as x and y � 16 when x � 5, find y when x � 20. (Lesson 13-6) 4

50. If y varies inversely as x and y � 2 when x � 25, find x when y � 40. (Lesson 13-6) 1.25

Graph each system of inequalities. (Lesson 13-7) 51–53. See additional answers.

51. x2 � y2 � 9 52. x2 � y2 � 1 53. y � x2

x2 � 4y2 � 16 x2 � y2 � 16 y � �x � 2

Graph each function. State the y -intercept. (Lesson 13-8)

54. y � ��15

��x

1 55. y � 5x � 4 �3 56. y � 4 2x 4

57. POPULATION The population of Canada has been increasing by an annualrate of 0.3%. The population of Canada was 32,207,000 in 2003. Predict itspopulation in 2018. (Lesson 13-8)

y

xO


(x � 1)2 � (y � 3)2 � 144 (x � 5)2 � (y � 4)2 � 36 (x � 4)2 � (y � 1)2 � 4

center (0, 2), radius 4 center (�1, �5), radius 1 center (7, 4), radius 7

focus (0, 5), directrix y � � 5 focus (0, �2), directrix y � 2 focus (0, 1), directrix y � �1

54–56. See additional answers for graphs.

about 33,687,150

Work with a partner.

1. In the function at the right, the variable x is an exponent of the base 2, and the value of 2x is to be determined. Copy and complete the table.

2. In the function at the right, the variable x is given as the power of 2, and the exponent yis to be determined. Copy and complete the table.

3. Compare the two functions. The functions are inverses of each other.

BUILD UNDERSTANDING

In the second function, x � 2y, the exponent y is called the logarithm base 2 of x.This function is written log2 x � y and is read “the log base 2 of x is equal to y.”The logarithm corresponds to the exponent.

E x a m p l e 1

Write each equation in logarithmic form.

a. 80 � 1 b. 2�4 � �116�

Solutiona. 80 � 1 log8 1 � 0 b. 2�4 � �

116� log2 �

116� � �4


13-9 Logarithmic FunctionsGoals ■ Evaluate logarithmic expressions.

■ Solve logarithmic equations.

Applications Sound, Chemistry, Earthquakes

x 2x � y y

�1 2�1 � y �12

�

0 20 � y 11 21 � y 22 22 � y 43 23 � y 84 24 � y 165 25 � y 32

x x � 2y y

�12

� �12

� � 2y �1

1 1 � 2y 02 2 � 2y 14 4 � 2y 28 8 � 2y 3

16 16 � 2y 432 32 � 2y 5

Exponential Function Logarithmic Functionn � bp p � logb n

exponent or logarithmbase

number

Animationmathmatters3.com


E x a m p l e 2

Write each equation in exponential form.

a. log9 81 � �2 b. log10 10,000 � 4

Solutiona. log9 �

811� � �2 �

811� � 9�2 b. log10 10,000 � 4 10,000 � 104

You can use the definition of logarithm to find the value of a logarithmic expression.

E x a m p l e 3

Evaluate log5 125.

Solutionlog5 125 � y Let the logarithm equal y.

125 � 5y Rewrite the equation using the definition of logarithm.

53 � 5y 53 � 125

Since 53 � 5y, y must equal 3.

A logarithmic equation is an equation that contains one or more logarithms. You can use the definition of a logarithm to help you solve logarithmic equations.

E x a m p l e 4

Solve log4 m � �5.

Solutionlog4 m � �5

m � 4�5 Rewrite the equation using the definition of logarithm.

m � �415� Use the definition of negative exponents.

m � �10

124� Simplify.

The property of equality for logarithmic equations states that if logb x � logb y, then x � y.

E x a m p l e 5

Solve log5 (x � 5) � log5 7.

Solutionlog5 (x � 5) � log5 7

x � 5 � 7 Use the property of equality for logarithmic equations.

x � 5 � 5 � 7 � 5 Add 5 to each side.

x � 12 Simplify.

Lesson 13-9 Logarithmic Functions 601mathmatters3.com/extra_examples


TRY THESE EXERCISES


1. 54 � 625 log5 625 � 4 2. 7�2 � �419� log7 �

419� � �2 3. 2�9 � �

5112� log2 �

5112� ��9


4. log3 81 � 4 81 � 34 5. log6 216 � 3 216 � 63 6. log5 �215� � �2 �

215� � 5�2

Evaluate each expression.

7. log2 64 6 8. log3 �217� �3 9. log10 1,000,000 6

Solve each equation.

10. log3 k � 6 729 11. log5 (2a � 3) � log5 21 9 12. log2 (3d � 5) � log2 (d � 7) 6



13. 85 � 32,768 14. ��13

��4

� �811� log �

811� � 4 15. 4�3 � �

614� log4 �

614� � �3

16. ��25

��3

� �1

825� log �

1825� � 3 17. ��

19

��3

� 729 log 729 � �3 18. 203 � 8000 log20 8000 � 3


19. log7 1 � 0 1 � 70 20. log4 64 � 3 64 � 43 21. log10 �1

100� = �2 �

1100� � 10�2

22. log3 729 � 6 729 � 36 23. log3 �2

143� � �5 �

2143� � 3�5 24. log 4 � �2 4 � ��

12

��2

Evaluate each expression.

25. log2 8 3 26. log12 1 0 27. log �26

74� 3

28. log4 �2

156� �4 29. log 64 �2 30. log10 0.00001 �5


31. log7 s � 5 16,807 32. log2 t � �5 �312� 33. log y � �6 46,656

34. log3 (4 � y) � log3 (2y) 4 35. log7 d � log7 (3d � 10) 5

36. log10 (4w � 15) � log10 35 5 37. log3 (3x � 6) � log3 (2x � 1) 7

38. log6 (3r � 1) � log6 (2r � 4) 5 39. log9 (5p � 1) � log9 (3p � 7) 4

40. YOU MAKE THE CALL Shane says that the value of log4 2 is 2. Do youagree? If not, why not? No; 42 � 16 not 2. The answer is �

12

�.

41. SOUND An equation for loudness L, in decibels, is L � 10 log10 R, where R isthe relative intensity of the sound. Find the relative intensity of a fireworksdisplay with a loudness of 150 decibels. 1015 or 1,000,000,000,000,000

1�6

1�8

3�4

1�2

1�9

2�5

1�3


log8 32,768 � 5

42. CHEMISTRY The pH of a solution is a measure of its acidity and is writtenas a logarithm with base 10. A low pH indicates an acidic solution. Neutralwater has a pH of 7. A substance has a pH of 4. How many times as acidic isthe substance as water? 103 or 1000 times

43. EARTHQUAKES The magnitude of an earthquake is measured on alogarithmic scale called the Richter scale. The magnitude is given by M � log10 x, where x represents the amplitude of the seismic wave. How many more times as great is the amplitude caused by an earthquakewith a Richter scale rating of 6 as an aftershock with a Richter scale rating of 4? 102 or 100 times


For Exercises 44–46, use the same coordinate plane for each graph. Use thetables in the opening activity on page 600 to help graph the functions.

44. Graph seven ordered pairs that satisfy the function y � 2x. Draw a smoothcurve through the points. Label the graph y � 2x.

45. Graph seven ordered pairs that satisfy the function x � 2y. Draw a smoothcurve through the points. Label the y � log2 x.

46. Graph x � y. Describe the relationship among the three graphs.

47. Compare the domain and range of the functions y � 2x and y � log2 x.

48. Graph the function y � 3x. Without using a table of values, graph y � log3 xon the same coordinate plane. See additional answers.


Simplify. (Lesson 11-4)

49. (2a � 4)(a � 1) 2a2 � 2a � 4 50. (3v � 1)(2v � 2) 6v2 � 8v � 2 51. (4g � 5)(4g � 5)

There are 9 pennies, 7 dimes, and 5 nickels in an antique coin collection.Suppose two coins are to be selected at random from the collection withoutreplacing the first one. Find the probability of each event. (Lesson 9-4)

52. P(a penny, then a dime) �230� 53. P(two nickels) �

211� 54. P(two dimes) �

110�

Solve each system of equations. (Lesson 6-6)

55. 2r � 3s � 11 56. 4c � 2d � 10 57. 4a � 2b � 15

2r � 2s � 6 (4, �1) c � 3d � 10 (1, 3) 2a � 2b � 7 �4, ��12

��

Given f (x ) = 3x � 10, evaluate each function. (Lesson 2-2)

58. f(2) �4 59. f(10) 20 60. f(�3) �19

Lesson 13-9 Logarithmic Functions 603

44–46. See additional answers for graphs.

The graphs of y � 2x and y � log2 x are reflections of each other over the line x � y.


16g2 � 25




Chapter 13 ReviewVOCABULARY

Choose the word from the list at the right that completes each statement below.

1. The set of all points equidistant from a fixed point called a focus and a fixed line called a directrix is a(n) ___?__.

2. A relationship in which one variable increases as the othervariable increases is a(n) ___?__.

3. A relationship in which one variable decreases as the othervariable increases is a(n) ___?__.

4. The equation (x � h)2 � (y � k)2 � r where r � 0 is the standard form of a(n) ___?__.

5. When a plane intersects right circular cones, a(n) ___?__ is formed.

6. Exponential and ___?__ functions are inverses of each other.

7. If a quantity decreases by a fixed rate each time period, there is exponential ___?__.

8. A set of points such that the sum of the distances from two fixed points called foci is always the same is a(n) ___?__.

9. A set of points such that the difference between the distances from two fixed points called foci is always the same is a(n) ___?__.

10. A line that a graph approaches, but never meets is a(n) ___?__.

LESSON 13-1 The Standard Equation of a Circle, p. 562

� The equation for a circle with its center at the origin and with radius r is x 2 � y 2 � r 2, r � 0.

� The standard equation for a circle with its center located at the point (h, k)with radius r is (x � h)2 � (y � k)2 � r 2, r � 0.


11. radius 8 12. radius 4 13. radius 6center (0, 0) center (2, 3) center (5, 0)

Find the radius and center of each circle.

14. x 2 � y 2 � 25 15. x 2 � (y � 3)2� 9 16. (x � 9)2� (y � 4)2 � 21

LESSON 13-2 More on Parabolas, p. 566

� When the focus (0, a) is on the y-axis and the directrix is y � �a, the standardequation for a parabola is x 2 � 4ay.

Find the focus and directrix of each parabola.

17. x 2 � 20y 18. �40y � 5x 2 19. 12x 2 � 48y � 0

a. asymptote

b. circle

c. conic section

d. decay

e. directrix

f. direct variation

g. ellipse

h. growth

i. hyperbola

j. inverse variation

k. logarithmic

l. parabola

Chapter 13 Review 605


20. Focus (0, 5) 21. Focus (0, �4) 22. Focus �0, �12

��LESSON 13-3 Problem Solving Skills: Visual Thinking, p. 572

� You can visualize the conic section formed by a plane intersecting a cone ordouble cone.

23. Describe the intersection between a plane and a double cone that producesa circle.

24. Describe the intersection between a plane and a double cone that produces a hyperbola.

LESSON 13-4 Ellipses and Hyperbolas, p. 574

� The standard form for the equation of an ellipse with its center at the origin

and foci on the x-axis is � � 1.

� The standard form for the equation of a hyperbola that is symmetric about the

origin and has foci on the x-axis is � � 1.

Find an equation for each figure.

25. an ellipse with foci (8, 0) and (�8, 0) and x-intercepts (10, 0) and (�10, 0)

26. a hyperbola with center (0, 0) and foci on the x-axis if a � 4 and b � 7

LESSON 13-5 Direct Variation, p. 580

� Equations in which one variable increases as the other variable increases canbe expressed as y � kx, where k is a positive constant and x � 0.

� Direct square variation is shown by the equation y � kx 2.

27. If y varies directly as x and y � 75 when x � 7.5, find y when x � 5.

28. If y varies directly as x 2 and y � 51.2 when x � 4, find y when x � 9.

29. Let y vary directly as the square of x. If y � 45 when x � 3, find y when x � 8.

LESSON 13-6 Inverse Variation, p. 584

� Equations in which one variable decreases as the other variable increases can

be expressed as y � �kx

�, where k is a nonzero constant and x � 0.

� Inverse square variation is shown by the equation y � , or x2y � k.



32. Let y vary inversely as x. If y � 6.5 when x � 3, find y when x � 4.

33. Let y vary inversely as the square of x. If y � 40 when x � 9, find y when x � 6.

k�x 2

y 2

�b 2

x 2

�a 2

y 2

�b 2

x 2

�a 2


LESSON 13-7 Quadratic Inequalities, p. 590

� Substitute coordinates into the equation for a quadratic inequality to locateregions in the solution set.

� Systems of inequalities can be solved by finding the intersections of their graphs.


34. x 2 � y 2 � 49 35. y � 2x 2 � x � 2 36. 9x 2 � 36y 2 � 36

LESSON 13-8 Exponential Functions, p. 594

� A function where the variable is an exponent is an exponential function.

� A quantity that increases or decreases by a fixed rate each time period is calledexponential growth or exponential decay, respectively.

Graph each function. State the y-intercept.

37. y � ��18

��x

38. y � 4��13

��x

39. y � 5(2x) � 4

40. POPULATION In 2002, the population of South Carolina was about 4,107,000. If itcontinues to grow at a rate of 1.1% per year, predict the population in 2012.

LESSON 13-9 Logarithmic Functions, p. 600

� The exponential function n � bp can be written as the logarithmic functionp � logb n.

� A logarithmic equation is an equation that contains one or more logarithms.

� The property of equality for logarithmic equations states that if logb x � logb y,then x � y.


41. 85 � 32,768 42. ��25

��3

� �1

825� 43. 172 � 289

44. 222 � 484 45. ��15

��4

� �6

125� 46. 76 � 117,649


47. log5 1 � t 48. log4 a � 5 49. log7 m � �3

50. log6 216 � x 51. logy 25 � 2 52. log4 z � �4

CHAPTER INVESTIGATIONEXTENSION Write a report to summarize your work. Be sure to include theresults of your research, your map, and a description of anything you learnedduring your class discussion of your work.

Chapter 13 Assessment

Chapter 13 Assessment 607mathmatters3.com/chapter_assessment


1. radius 6 2. radius 10 3. radius 13center (0, 0) center (1, �5) center (�1, 7)


4. x 2 � y 2 � 21 5. (x � 4)2 � y 2 � 11 6. (x � 7)2 � (y � 6)2 � 225

Find the focus and directrix for each equation.

7. x 2 � 32y 8. x 2 � �24y 9. 32y � 4x 2 � 0


10. Focus (0, 8) 11. Focus (0, �2) 12. Focus �0, ��14

��Find an equation for each figure.

13. an ellipse with foci (4, 0) and (�4, 0) and x-intercepts (5, 0) and (�5, 0)

14. a hyperbola with center (0, 0) and foci on the x-axis if a � 5 and b � 11

Solve.

15. If y varies directly as x and y � 36 when x � 15, what is y when x � 19?

16. If y varies inversely as x and y � 72 when x � 9, what is y when x � 6?

17. If y varies inversely as x and y � 144 when x � 6, what is y when x � 4?

18. If y varies directly as x and y � 360 when x � 12, what is y when x � 18?

19. The total area of a picture and its frame is 456 in.2. The picture is 21 in. long and 16 in. wide. What is the width of the frame?


20. (x � 1)2 � (y � 6)2 � 64 21. y � x 2 � 4x 22. 9x 2 � 25y 2 � 225

23. The amount of time a projectile is in the air after launch can be found by solving vt � 16t 2 � 0 (v � initial upward velocity, t � time). How long is a baseball in the air if it is thrown with an upward velocity of 64 ft/sec?

24. Write the equation of a circle having center at (2, �1) and radius of 3.

25. If y varies directly as x 2 and y � 112 when x � 4, find y when x � 5.

26. INVESTMENTS Determine the amount of an investment if $10,000 is invested at aninterest rate of 4.5% each year for 6 yr.

27. Write 105 � 100,000 in logarithmic form.

28. What is the value of log3 243?

http://mathmatters3.com/chapter_assessment


Standardized Test Practice6. Which is the simplest radical form of the

product of (7�3� and (2�21�)? (Lesson 10-1)

18�6� 42�7�14�63� 9�24�

7. Find the value of x in the figure. (Lesson 10-6)

�175� �

251�

4 �335�

8. If z2 � 10z � 3 � 0, which is the value of z?(Lesson 12-5)

�5 � 2�22��5 � �22�5 � �22�5 � 2�22�

9. Which equation represents a graph that is aparabola? (Lesson 13-2)

(x � 3)2 � (y � 2)2 � 100

x2 � 4y2 � 4

x2 � 20y � 0

x2 � 25y2 � 100

10. Which graph represents the system ofinequalities? (Lesson 13-7)

4x2 � 9y2 � 36

x � y � 1 � 0

2

2

y

x

D

2

2

y

x

C

2

2

y

x

B

2

2

y

x

A

D

C

B

A

D

C

B

A

DC

BA 75

3

x

DC

BA

Part 1 Multiple Choice

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

1. The sum of the measures of the complement ofan angle and the measure of its supplement is144°. Which is the measure of the angle?(Lesson 3-2)

18° 27°

63° 72°

2. Line p is parallel to line q. Which is the value ofc � a? (Lesson 3-4)

0° 100°

120° 150°

3. Which is the longest segment in the figure?(Lesson 4-6)

AB��BC��CCD��DA��

4. Choose the equation of a line parallel to thegraph of y � 3x � 4. (Lesson 6-2)

y � ��13

�x � 4

y � �3x � 4

y � �x � 1

y � 3x � 5

5. Drawing a card at random from a standard deckof cards, which is the probability that the cardis a diamond or a face card? (Lesson 9-3)

�532� �

133�

�12

16� �

25

52�DC

BA

D

C

B

A

D

C

B

AA

54�

50�

61�

64�

B

C

D

DC

BA

ac

p

40

q

DC

BA

Test-Taking TipQuestion 21You cannot write mixed numbers, such as 2�

12

�, on an answer grid. Answers such as these need to be written as improperfractions, such as 5/2, or as decimals, such as 2.5. Choose the method that you like best, so that you will avoid makingunnecessary mistakes.

Chapter 13 Standardized Test Practice 609mathmatters3.com/standardized_test

Preparing for Standardized TestsFor test-taking strategies and morepractice, see pages 709–724.

Part 2 Short Response/Grid In

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

11. What is the value of � � �(x � 1) � if x � �4?(Lesson 1-2)

12. Solve 9(x � 4) � 2x � 19 � 3(x � 6).(Lesson 2-5)

13. What is the volume of the pyramid?(Lesson 5-7)

14. What is the point of intersection of the graphsof x � 2y � 10 and 2x � y � 5? (Lesson 6-4)

15. Rectangle ABCD is similar to rectangle EFGH. If AB � 6, BC � 7, and EF � 9, what is the perimeter of rectangle EFGH?(Lesson 7-2)

16. In �XYZ, AB��and CD��are parallel to XY��.If YB � 2, BD � 3, DZ � 4, and AC � 6, find AZ.(Lesson 7-6)

17. Point A(2, 5) is rotated 90° counterclockwiseabout the origin. What are the coordinates ofthe image? (Lesson 8-2)

18. The measure of �ABCis 56°. What is the measure of �AOC?(Lesson 10-4)

19. What is the product of (2x � 3y) and (4x � y)?(Lesson 11-4)

56�

O

B

A

C

A B

X Y

DC

Z

6 ft

4 ft 4 ft

20. What number must be added to a2 � 14ato make it a perfect square trinomial?(Lesson 12-4)

21. If y varies inversely as x, and one pair of values is y � 14 and x � 5, find y when x � 20.(Lesson 13-6)

22. Find log2 �18

�. (Lesson 13-9)

Part 3 Extended Response

Record your answers on a sheet of paper. Showyour work.

23. Draw the graph of the equation 16x2 � 25y2 � 400. Plot and label the x-intercepts, the y-intercepts, and the foci.(Lesson 13-4)

24. A friend wants to enroll for cellular phoneservice. Three different plans are available.(Lesson 13-5)Plan 1 charges $0.59 per minute.Plan 2 charges a monthly fee of $10, plus

$0.39 per minute.Plan 3 charges a monthly fee of $59.95.a. For each plan, write an equation that

represents the monthly cost C for mnumber of minutes per month.

b. Which plan(s) represent a direct variation?

c. Your friend expects to use 100 min permonth. In which plan do you think thatyour friend should enroll? Explain.

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chapter 13: advanced functions and relations

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