8-2 the reciprocal function - mr. gongora's website · the reciprocal function family 8-2...

Lesson 8-2 The Reciprocal Function Family 507

The Reciprocal Function Family

8-2

Objectives To graph reciprocal functionsTo graph translations of reciprocal functions

Functions that model inverse variation have the form f (x) 5 ax, where x 2 0. Th ey

belong to a family whose parent is the reciprocal function f (x) 5 1x, where x 2 0.

Essential Understanding Transformations of the parent reciprocal function include stretches, compressions (or shrinks), refl ections, and horizontal and vertical translations.

Key Concept General Form of the Reciprocal Function Family

Th e general form of a member of the reciprocal function family is y 5 a

x 2 h 1 k, where x 2 h.

Th e inverse variation functions, y 5 ax, are stretches, shrinks,

and refl ections of the parent reciprocal function, depending on the value of a.

Th e graph of the parent reciprocal function y 5 1x is shown at

the right.

4

O 424

2

4

x

y

For a class party, the students will share the cost for the hall rental. Each student will also have to pay $8 for food. The cost of the hall rental is already graphed. What effect does the food cost have on the graph? Explain your reasoning.

Lesson Vocabulary

• reciprocal function

• branch

V•

V•

Dynamic ActivityGraphing Translations of Inverse Variations T

AC T I V I T I

E S

AAAAAAAAC

ACC

I ESSSSSSSS

DYNAMIC

.

y

O 20 40 60 80 100

21

345678

x

Number of students

Cost of Class Party

Cost

per

stu

dent

($)

(40, 6)

(60, 4)(80, 3)

(100, 2.4)

Content StandardsF.BF.3 Identify the effect on the graph of replacing f (x)by f(x) 1 k, kf(x), f(kx), and f(x 1 k) for specifi c values of k . . .

A.CED.2 Create equations in two or more variables to represent relationships between quantities . . .

Also A.APR.1

0507_hsm11a2se_cc_0802.indd 5070507_hsm11a2se_cc_0802.indd 507 4/21/11 5:46:18 PM4/21/11 5:46:18 PM

Problem 1

Got It?

508 Chapter 8 Rational Functions

Graphing an Inverse Variation Function

What is the graph of y 5 8x, x u 0? Identify the x- and y-intercepts and the asymptotes

of the graph. Also, state the domain and range of the function.

Step 1 Make a table of values that Step 2 Graph the points. includes positive and negative values of x.

Step 3 Connect the points with a smooth curve. x cannot be zero, so there is no y-intercept. Th e numerator is never zero, so y is never 0. Th ere is no x-intercept.

Th e x-axis is a horizontal asymptote. Th e y-axis is a vertical asymptote.

Knowing the asymptotes provides you with the basic shape of the graph.

Th e domain is the set of all real numbers except x 5 0.

Th e range is the set of all real numbers except y 5 0.

1. a. What is the graph of y 5 12x ? Identify the x- and y-intercepts and the

asymptotes of the graph. Also, state the domain and range of the function.

b. Reasoning Would the function y 5 6x have the same domain and range

as y 5 8x or y 5 12

x ? Explain.

Each part of the graph of a reciprocal function is a branch. Th e branches of the parent function y 5 1

x are in Quadrants I and III. Stretches and compressions of the parent function remain in the same quadrants. Refl ections are in Quadrants II and IV.

y

8

16

O 816

16

16x

What values should you choose for x?Choose values of x that divide nicely into 8. Make a table of points that are easy to graph.

x y

16

8

4

2

1

1

2

4

8

12

12 16

x y

16

8

4

2

1

1

2

4

8

12

1216

y

8

16

O 816

16

16xx

Notice how the y-valuesget closer to zero as the absolute values of x get larger.

O 88

66

16

The absolute values ofy get very large as x approaches zero.


Problem 2

Got It?


Identifying Reciprocal Function Transformations

For each given value of a, how do the graphs of y 5 1x and y 5 a

x compare? What is the eff ect of a on the graph?

A a 5 6

Th e graph (in red) of y 5 6x is a

stretch of the graph of y 5 1x

(in black) by the factor 6.

B a 5 0.25

Th e graph (in blue) of y 5 0.25x is a

shrink of the graph of y 5 1x

(in black) by the factor 14.

C a 5 26

Th e graph of y 5 26x is the stretch

by the factor 6 in part A followed by a refl ection across the x-axis.

2. For each given value of a, how do the graphs of y 5 1x and y 5 a

x compare? What is the eff ect of a on the graph?

a. a 5 12 b. a 5 2 c. a 5 2

12

y

2

4

O 2 4x

y 6x

O 22 44x

y 0.25x

How does the negative sign affect the graph?The y-values have signs that are opposite those in part A. The graph in A refl ects across the x-axis.

y 1x

y 6x

You can translate any reciprocal function horizontally or vertically just as you can other functions.

Key Concept The Reciprocal Function Family

Parent function y 5 1x, x 2 0

Stretch ( u a u . 1)

Shrink (0 , u a u , 1)

Reflection (a , 0) across x-axis

y 5 ax, x 2 0

Translation (horizontal by h; vertical by k)with vertical asymptote x 5 h horizontal y 5 1

x 2 h 1 k; x 2 hasymptote y 5 k

Combined y 5 ax 2 h 1 k; x 2 h

t


Problem 3

Problem 4

Got It?

Got It?


When you graph a translated reciprocal function, a good fi rst step is to draw the asymptotes.

Graphing a Translation

What is the graph of y 5 1x 1 1 2 2? Identify the domain and range.

Step 1 Draw the asymptotes (red).

For, y 5 1x 1 1 2 2, h 5 21 and k 5 22.

Th e vertical asymptote is x 5 21. Th e horizontal asymptote is y 5 22.

Step 2 Translate the graph of y 5 1x.

Th e graph of y 5 1x contains the points (1, 1) and (21, 21). Translate these

points 1 unit to the left and 2 units down to (0, 21) and (22, 23), respectively.Draw the branches through these points (blue).

Th e domain is the set of all real numbers except x 5 21. Th e range is the set of all real numbers except y 522.

3. What is the graph of y 5 1x 2 4 1 6? Identify the domain and range.

If you know the asymptotes of the graph of a reciprocal function and the value of a, you can write the equation of the function.

Writing the Equation of a Transformation

Multiple Choice Th is graph of a function is a translation of the graph of y 5 2

x. What is an equation for the function?

y 5 2x 1 3 1 4 y 5 2

x 2 3 1 4

y 5 2x 1 3 2 4 y 5 2

x 2 3 2 4

Th e asymptotes are x 5 23 and y 5 4. Th us h 5 23 and k 5 4.

y 5 ax 2 h 1 k Use the general form.

y 5 2x 2 (23)

1 4 Substitute for a, h, and k.

y 5 2x 1 3 1 4 Simplify.

Th e correct choice is A.

4. Th is graph of a function is a translation of the graph of y 5 2

x. What is an equation for the function?

y

448 8x

y

48 8x

4

8

O

448 8x

4

8

O

How do you fi nd the asymptotes?The asymptotes of y 5 1

x (the axes) translate 1 unit to the left and 2 units down.

How can you get started?Identify the asymptotes of the graph.


Problem 5

Got It?


Using a Reciprocal Function

Clubs Th e rowing club is renting a 57-passenger bus for a day trip. Th e cost of the bus is $750. Five passengers will be chaperones. If the students who attend share the bus cost equally, what function models the cost per student C with respect to the number of students n who attend? What is the domain of the function? How many students must ride the bus to make the cost per student no more than $20?

To share the cost equally, divide 750 by the number of students, n, who attend.

Th e function that models the cost per student is C 5 750n .

Th e bus has a capacity of 57 passengers and there will be 5 chaperones. Th e maximum number of students is 57 2 5 5 52.

Th e domain is the integers from 1 to 52.

Use a graphing calculator to solve the inequality 750n # 20. Let

Y1 5 750x and Y2 5 20.

Change the window dimensions to get a closer look at the graph. Use the intersect feature.

At least 38 students must ride the bus.

5. Th e junior class is renting a laser tag facility with a capacity of 325 people. Th e cost for the facility is $1200. Th e party must have 13 adult chaperones.

a. If every student who attends shares the facility cost equally, what function models the cost per student C with respect to the number of students n who attend? What is the domain of the function? How many students must attend to make the cost per student no more than $7.50?

b. Th e class wants to promote the event by giving away 30 spots to students in a drawing. How does the model change? Now how many paying students must attend so the cost for each is no more than $7.50?

• The bus holds 57 passengers.

• The bus costs $750. • Five riders are chaperones

who pay nothing for the bus.

• A function for the cost per student

• The number of students needed so that the cost does not exceed $20 per student

• Write a reciprocal function for the situation.

• Graph the function and solve an inequality using the $20 limit.

If x 37, the costwill be more than $20.Intersection

X = 37.5 Y = 20

For all values greater than or equalto 38, the cost is less than $20.

IntersX = 37

The number of peoplemust be a whole number.

Is the domain x K 52?No; the domain is the possible numbers of students, so only positive integers make sense.



Lesson Check

Practice and Problem-Solving Exercises

Graph each function. Identify the x- and y-intercepts and the asymptotes of the graph. Also, state the domain and the range of the function.

8. y 5 2x 9. y 5 15

x 10. y 5 23x 11. y 5 2

10x 12. y 5 10

x

Graphing Calculator Graph the equations y 5 1x and y 5 a

x using the given value of a. Th en identify the eff ect of a on the graph.

13. a 5 2 14. a 5 24 15. a 5 0.5 16. a 5 12 17. a 5 0.75

Sketch the asymptotes and the graph of each function. Identify the domain and range.

18. y 5 1x 2 3 19. y 5 22

x 2 3 20. y 5 1x 2 2 1 5 21. y 5 1

x 2 3 1 4

22. y 5 2x 1 6 2 1 23. y 5 10

x 1 1 2 8 24. y 5 1x 2 2 25. y 5 28

x 1 5 2 6

Write an equation for the translation of y 5 2x that has the given asymptotes.

26. x 5 0 and y 5 4 27. x 5 22 and y 5 3 28. x 5 4 and y 5 28

29. Construction Th e weight P in pounds that a beam can safely carry is inversely proportional to the distance D in feet between the supports of the beam. For a certain type of wooden beam, P 5 9200

D . What distance between supports is needed to carry 1200 lb?

30. Think About a Plan A high school decided to spend $750 on student academic achievement awards. At least 5 awards will be given, they should be equal in value, and each award should not be less than $50. Write and sketch a function that models the relationship between the number a of awards and the cost c of each award. What are the domain and range of the function?

• Which equation describes the relationship between a and c? • What information can you use to determine the domain and range?

PracticeA See Problem 1.

See Problem 2.

See Problem 3.

See Problem 4.

STEM See Problem 5.

ApplyB

Do you know HOW? 1. Graph the equation y 5 3

x.

Describe the transformation from the graph of y 5 1x to

the graph of the given function.

2. y 5 1x 1 5 3. y 5 24

x

4. What are the asymptotes of the graph of y 5 5

x 1 2 2 7?

Do you UNDERSTAND? 5. Vocabulary What transformation changes the graph

of y 5 1x into the graph of y 5 1

2x ?

6. Open Ended Write an equation of a stretch and a refl ection of the graph y 5 1

x across the x-axis.

7. Writing Explain how you can tell if a function y 5 ax

is a stretch or compression of the parent function y 5 1

x.

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

0507_hsm11a2se_cc_0802.indd 5120507_hsm11a2se_cc_0802.indd 512 5/11/11 4:43:58 AM5/11/11 4:43:58 AM


31. Open-Ended Write an equation for a horizontal translation of y 5 2x. Th en write

an equation for a vertical translation of y 5 2x. Identify the horizontal and vertical

asymptotes of the graph of each function.

Sketch the graph of each function.

32. xy 5 3 33. xy 1 5 5 0 34. 3xy 5 1 35. 5xy 5 2 36. 10xy 5 24

37. Writing Explain how knowing the asymptotes of a translation of y 5 1

x can help you graph the function. Include an example.

38. Multiple Choice Th e formula p 5 69.1a 1 2.3

models the relationship between atmospheric pressure p in inches of mercury and altitude a in miles.

Use the data shown with the photo. At which location does the model predict the pressure to be about 23.93 in. of mercury? (Hint: 1 mi 5 5280 ft.)

Sahara Desert

Kalahari Desert

Mt. Kilimanjaro

Vinson Massif

Graphing Calculator Graph each pair of functions. Find the approximate point(s) of intersection.

39. y 5 6x 2 2, y 5 6 40. y 5 2

1x 2 3 2 6, y 5 6.2 41. y 5 3

x 1 1, y 5 24

42. Reasoning How will the domain and the range of the parent function y 5 1x change

after the translation of its graph by 3 units up and by 5 units to the left?

43. a. Gasoline Mileage Suppose you drive an average of 10,000 miles each year. Your gasoline mileage (mi/gal) varies inversely with the number of gallons of gasoline you use each year. Write and graph a model for your average mileage m in terms of the gallons g of gasoline used.

b. After you begin driving on the highway more often, you use 50 gal less per year. Write and graph a new model to include this information.

c. Calculate your old and new mileage assuming that you originally used 400 gal of gasoline per year.

Reasoning Compare each pair of graphs and fi nd any points of intersection.

44. y 5 1x and y 5 P 1x P 45. y 5 1

x and y 5 1x2 46. y 5 P 1x P and y 5 1

x2

47. Find two reciprocal functions such that the minimum distance from the origin to the graph of each function is 4!2.

48. Write each equation in the form y 5 kx 2 b 1 c, and sketch the graph.

a. y 5 23x 2 6 b. y 5 1

2 2 4x c. y 5 3 2 xx 1 2 d. xy 2 y 5 1

ChallengeC

of e.

which ure

Sahara Desertaverage alt. 1500 ft

Kalahari Desertaverage alt. 3100 ftaverage alt. 310

Mt. Kilimanjaroalt. 19,340 ft

Vinson Massifalt. 16,680 ft



Mixed Review

Suppose that x and y vary inversely. Write a function that models each inverse variation and fi nd y when x 5 25.

54. x 5 2 when y 5 12 55. x 5 25 when y 5 2 56. x 5 12 when y 5 4

Without graphing, determine whether the function represents exponential growth or exponential decay. Th en fi nd the y-intercept.

57. y 5 3(4)x 58. y 5 0.1(2)x 59. y 5 5(0.8)x 60. y 5 3Q12R

x

Multiply.

61. (5!3 2 2)2 62. (4 1 2!3)(6 2 3!3) 63. (!3 1 !5)(!3 2 !5)

Get Ready! To prepare for Lesson 8-3, do Exercises 64–67.

Factor each expression.

64. x2 2 6x 1 8 65. x2 1 6x 2 27 66. 2x2 1 x 2 28 67. 2x2 2 19x 1 24

See Lesson 8-1.

See Lesson 7-1.

See Lesson 6-3.

See Lesson 4-4.

Standardized Test Prep

49. What is an equation for the translation of y 5 2x that has asymptotes at x 5 3 and

y 5 25?

y 5 2x 2 3 2 5 y 5 2

x 1 3 1 5 y 5 2x 1 5 2 3 y 5 2

x 2 5 1 3

50. Th e graph at the right shows which inequality?

y , 22.5x 1 5 22.5x 1 y , 5

2.5x 1 y $ 5 5x 1 y # 5

51. If p and q vary inversely, and p 5 10 when q 5 24, what is q when p 5 22?

20 45 2 45 220

52. Which equation represents the inverse of the graph at the right?

y 5 log3 x y 5 logx 3

x 5 log3 y x 5 logy 3

53. What is b if the graph of y 5 27bx includes the point (21, 81)?

SAT/ACT

x

y

O

8

84

8

48

4

4

x

y

9

3

6

O 24 2 4x

ShortResponse


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