Holt Algebra 1 12-2 Rational Functions 12-2 Rational Functions Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson

Holt Algebra 1

12-2 Rational Functions12-2 Rational Functions

Holt Algebra 1

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Algebra 1

12-2 Rational Functions

Warm Up

The inverse variation xy = 8 relates the constant speed x in mi/h to the time y in hours that it takes to travel 8 miles. Graph this inverse variation. Then use the graph to estimate how many hours it would take to travel 8 miles jogging at a speed of 4.5 mi/h.

Possible answer: 1 h 45 min

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Identify excluded values of rational functions.

Graph rational functions.

Objectives

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rational functionexcluded valuediscontinuous functionasymptote

Vocabulary

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A rational function is a function whose rule is a quotient of polynomials in which the denominator has a degree of at least 1. In other words, there must be a variable in the denominator. The inverse variations you studied in the previous lesson are a special type of rational function.

Rational functions:

Not rational functions:

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For any function involving x and y, an excluded value is any x-value that makes the function value y undefined. For a rational function, an excluded value is any value that makes the denominator equal to 0.

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Example 1: Identifying Excluded Values

Identify the excluded value for each rational function.

A.

x = 0 Set the denominator equal to 0.The excluded value is 0.

B.

x – 2 = 0

x = 2

Set the denominator equal to 0.Solve for x.

The excluded value is 2.

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Check It Out! Example 1


a.

x = 0 Set the denominator equal to 0.The excluded value is 0.

b.

x – 1 = 0

x = 1


The excluded value is 1.

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x + 4 = 0

x = –4


The excluded value is –4.

c.

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Most rational functions are discontinuous functions, meaning their graphs contain one or more jumps, breaks, or holes. This occurs at an excluded value.

One place that a graph of a rational function is discontinuous is at an asymptote. An asymptote is a line that a graph gets closer to as the absolute value of a variable increases. In the graph shown, both the x- and y-axes are asymptotes. The graphs of rational functions will get closer and closer to but nevertouch the asymptotes.

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For rational functions, vertical asymptotes will occur at excluded values.

Look at the graph of y = The denominator is 0 when x = 0 so 0 is an excluded value. This means there is a vertical asymptote at x = 0. Notice the horizontal asymptote at y = 0.

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Vertical lines are written in the form x = b, and horizontal lines are written in the form y = c.

Writing Math

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Look at the graph of y = Notice that the graph of the parent function y= has been translated 3 units right and there is a vertical asymptote at x = 3. The graph has also been translated 2 units up and there is a horizontal asymptote at y = 2.

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These translations lead to the following formulas for identifying asymptotes in rational functions.

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Example 2A: Identifying Asymptotes

Identify the asymptotes.

Step 2 Identify the asymptotes.

vertical: x = –7

horizontal: y = 0

Step 1 Write in y = form.

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Example 2B: Identifying Asymptotes


Step 1 Identify the vertical asymptote.

2x – 3 = 0

+3 +32x = 3

Find the excluded value. Set the denominator equal to 0.

Add 3 to both sides.

Solve for x. Is an excluded value.

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Example 2B Continued


Step 2 Identify the horizontal asymptote.

c = 8y = 8

Vertical asymptote: x = ; horizontal asymptote: y = 8

y = c

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Check It Out! Example 2a



x – 5 = 0

+5 +5x = 5


Add 5 to both sides.

Solve for x. 5 is an excluded value.x = 5

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c = 0y = 0 y = c

Check It Out! Example 2a Continued

Vertical asymptote: x = 5; horizontal asymptote: y = 0

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Check It Out! Example 2b



4x + 16 = 0

–16 –164x = –16


Subtract 16 from both sides.

Solve for x. –4 is an excluded value.x = –4

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Check It Out! Example 2b Continued



c = 5y = 5

Vertical asymptote: x = –4; horizontal asymptote: y = 5

y = c

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Check It Out! Example 2c



x + 77 = 0

–77 –77x = –77


Subtract 77 from both sides.

Solve for x. –77 is an excluded value.

x = –77

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Check It Out! Example 2c Continued



c = –15y = –15

Vertical asymptote: x = –77; horizontal asymptote: y = –15

y = c

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To graph a rational function in the form y =when a = 1, you can graph the asymptotes and then translate the parent function y = . However, if a ≠ 1, the graph is not a translation of the parent function. In this case, you can use the asymptotes and a table of values.

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Example 3A: Graphing Rational Functions Using Asymptotes

Graph the function.

Since the numerator is not 1, use the asymptotes and a table of values.

Step 1 Identify the vertical and horizontal asymptotes.

vertical: x = 3

horizontal: y = 0

Use x = b. x – 3 = 0, so b = 3.

Use y = c. c = 0

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Example 3A Continued

Step 2 Graph the asymptotes using dashed lines.

Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote.

y

654210x

x = 3

y = 0

Step 4 Plot the points and connect them with smooth curves. The curves will get very close to the asymptotes, but will not touch them.

●●●

●●●

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Example 3B: Graphing Rational Functions Using Asymptotes

Graph the function.

Step 1 Since the numerator is 1, use the asymptotes and translate y = .

vertical: x = –4

horizontal: y = –2

Use x = b. b = –4

Use y = c. c = –2

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Example 3B Continued

Step 3 Draw smooth curves to show the translation.

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Check It Out! Example 3a

Graph each function.

Step 1 Since the numerator is 1, use the asymptotes and translate y = .

vertical: x = –7

horizontal: y = 3

Use x = b. b = –7

Use y = c. c = 3

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Check It Out! Example 3a Continued



Step 3 Draw smooth curves to show the translation.

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Since the numerator is not 1, use the asymptotes and a table of values.


vertical: x = 3

horizontal: y = 2

Use x = b. x – 3 = 0, so b = 3.

Use y = c. c = 2

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Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote.

y

654210x

4

Step 4 Plot the points and connect them with smooth curves. The curves will get very close to the asymptotes, but will not touch them.

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Example 4: Application

Your club has $75 with which to purchase snacks to sell at an afterschool game. The number of snacks y that you can buy, if the average price of the snacks is x-dollars, is given by y =

a. Describe the reasonable domain and range values.

Both the number of snacks purchased and their cost will be positive values so nonnegative values are reasonable for both domain and range.

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Example 4 Continued

b. Graph the function.


vertical: x = 0

horizontal: y = 0

Use x = b. b = 0

Use y = c. c = 0

Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the x- and y-axes.

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Example 4 Continued

Step 3 Since the domain is restricted to nonnegative values, only choose x-values on the right side of the vertical asymptote.

Number of snacks 2 4 6 8Cost of snacks($) 37.5 18.75 12.5 9.38

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Example 4 Continued

Step 4 Plot the points and connect them with a smooth curve.

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a. Describe the reasonable domain and range values.

The domain would be all values greater than 0 up to $500 dollars and the range would be all natural numbers greater than 10.

A librarian has a budget of $500 to buy copies of a software program. She will receive 10 free copies when she sets up an account with the supplier. The number of copies y of the program that she can buy is given by

y = + 10, where x is the price per copy.

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Check It Out! Example 4 Continued



vertical: x = 0

horizontal: y = 10

Use x = b. b = 0

Use y = c. c = 10

Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the x- and y-axes.

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Step 3 Since the domain is restricted to nonnegative values, only choose x-values on the right side of the vertical asymptote.


Number of copies 20 40 60 80Price ($) 35 23 18 16

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Step 4 Plot the points and connect them with a smooth curve.


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The table shows some of the properties of the four families of functions you have studied and their graphs.

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Lesson Quiz: Part I

Identify the exceeded value for each rational function.

1. 2.

3. Identify the asymptotes of and then graph the function.

x = –4; y = 0

0 5

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Lesson Quiz: Part II

You have $ 100 to spend on CDs. A CD club advertises 6 free CDs for anyone who becomes a member. The number of CDs y

that you can receive is given by y = , where x is the average price per CD.

a. Describe the reasonable domain and range values.D: x > 0

R: natural numbers > 6

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Lesson Quiz: Part III


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Holt Algebra 1 12-2 Rational Functions 12-2 Rational Functions Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson