NAME
DATE
11-2
PERIOD
Study Guide and Intervention
Rational Functions
10
The function y = −
x is an example of a rational function.
Because division by zero is undefined, any value of a variable that results in a denominator
of zero must be excluded from the domain of that variable. These are called excluded
values of the rational function.
Identify Excluded Values
Example
State the excluded value for each function.
3
a. y = −
x
The denominator cannot equal zero.
The excluded value is x = 0.
4
b. y = −
x-5
x-5=0
Set the denominator equal to 0.
x=5
Add 5 to each side.
The excluded value is x = 5.
State the excluded value for each function.
2
1. y = −
x x = 0
1
2. y = −
x=4
x-3
3. y = −
x = -1
4
4. y = −
x=2
x
5. y = −
x=2
5
6. y = - −
x=0
3x - 2
7. y = −
x = -3
x-1
8. y = −
x = -2
9. y = −
x=0
x
x-2
2x - 4
x+3
3x
x+1
5x + 10
x-7
10. y = −
x = -4
x-5
11. y = −
x=0
x-2
12. y = −
x = -11
7
13. y = −
x = -7
3x - 4
14. y = −
x = -4
x
15. y = −
x=5
2x + 8
3x + 21
6x
x + 11
x+4
7x - 35
16. DINING Mya and her friends are eating at a restaurant. The total bill of $36 is split
36
among x friends. The amount each person pays y is given by y = −
x , where x is the
number of people. Graph the function.
36
32
Bill per Person ($)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x+1
x-4
28
24
20
16
12
8
4
0
1
2
3
4
5
6
7
8
Number of People
Chapter 11
11
Glencoe Algebra 1
Lesson 11-2
Exercises
NAME
DATE
11-2
Study Guide and Intervention
PERIOD
(continued)
Rational Functions
Identify and Use Asymptotes
Because excluded vales are undefined, they affect
the graph of the function. An asymptote is a line that the graph of a function approaches.
a
+ c has a vertical asymptote at the x-value that
A rational function in the form y = −
x-b
makes the denominator equal zero, x = b. It has a horizontal asymptote at y = c.
1
Identify the asymptotes of y = −
+ 2 . Then graph the function.
Example
x-1
Step 1 Identify and graph the asymptotes using dashed lines.
vertical asymptote: x = 1
horizontal asymptote: y = 2
Step 2 Make a table of values and plot the points.
Then connect them.
x
–1
0
2
3
y
1.5
1
3
2.5
y
y =2
x
0
1
y = x-1 + 2
x =1
Exercises
Identify the asymptotes of each function. Then graph the function.
-2
2. y = −
x x = 0; y = 0
x
2
4. y = −
x - 3 x = 0; y = 3
x
Chapter 11
x
x
0
0
2
5. y = −
x = 1; y = 0
-2
6. y = −
x = 3; y = 0
y
y
x+1
y
0
y
y
y
0
4
3. y = −
x + 1 x = 0; y = 1
0
12
x
x-3
0
x
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
1. y = −
x x = 0; y = 0