1. A rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = Its graph is a hyperbola, which has two separate branches. You will learn more about hyperbolas in Chapter 10. 1 x

2. Like logarithmic and exponential functions, rational functions may have asymptotes. The function f(x) = has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. 1 x

3. The rational function f(x) = can be transformed by using methods similar to those used to transform other types of functions. 1 x

4. Example 1: Transforming Rational Functions
Using the graph of f(x) = as a guide, describe the transformation and graph each function. 1
x + 2 1 x
– 3
A. g(x) =
B. g(x) =
Because h = –2, translate f 2 units left.
Because k = –3, translate f 3 units down.

5. Check It Out! Example 1
Using the graph of f(x) = as a guide, describe the transformation and graph each function. 1 x 1
x + 4 1 x
+ 1
a. g(x) =
b. g(x) =
Because k = 1, translate f 1 unit up.
Because h = –4, translate f 4 units left.

6. The values of h and k affect the locations of the asymptotes, the domain, and the range of rational functions whose graphs are hyperbolas.

7. Example 2: Determining Properties of Hyperbolas
Identify the asymptotes, domain, and range of the function g(x) = – 2. 1
x + 3 1
x – (–3)
g(x) = – 2
h = –3, k = –2.
Vertical asymptote: x = –3
The value of h is –3.
Domain: {x|x ≠ –3}
Horizontal asymptote: y = –2
The value of k is –2.
Range: {y|y ≠ –2}
Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = –3 and y = –2.

8. Check It Out! Example 2
Identify the asymptotes, domain, and range of the function g(x) = – 5. 1
x – 3 1
x – (3)
g(x) = – 5
h = 3, k = –5.
Vertical asymptote: x = 3
The value of h is 3.
Domain: {x|x ≠ 3}
Horizontal asymptote: y = –5
The value of k is –5.
Range: {y|y ≠ –5}
Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = 3 and y = –5.

9. A discontinuous function is a function whose graph has one or more gaps or breaks. The hyperbola graphed in Example 2 and many other rational functions are discontinuous functions.
A continuous function is a function whose graph has no gaps or breaks. The functions you have studied before this, including linear, quadratic, polynomial, exponential, and logarithmic functions, are continuous functions.

10. The graphs of some rational functions are not hyperbolas
The graphs of some rational functions are not hyperbolas. Consider the rational function f(x) = and its graph.
(x – 3)(x + 2)
x + 1
The numerator of this function is 0 when x = 3
or x = –2. Therefore, the function has x-intercepts at –2 and 3. The denominator of this function is 0 when x = –1. As a result, the graph of the function has a vertical asymptote at the line x = –1.

12. Some rational functions, including those whose graphs are hyperbolas, have a horizontal asymptote. The existence and location of a horizontal asymptote depends on the degrees of the polynomials that make up the rational function.
Note that the graph of a rational function can sometimes cross a horizontal asymptote. However, the graph will approach the asymptote when |x| is large.

14. Example 4A: Graphing Rational Functions with Vertical and Horizontal Asymptotes
Identify the zeros and asymptotes of the function. Then graph.
x2 – 3x – 4 x
f(x) =
x2 – 3x – 4 x
f(x) =
Factor the numerator.
The numerator is 0 when x = 4 or x = –1.
Zeros: 4 and –1
The denominator is 0 when x = 0.
Vertical asymptote: x = 0
Horizontal asymptote: none
Degree of p > degree of q.

15. Identify the zeros and asymptotes of the function. Then graph.
Example 4A Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph with a graphing calculator or by using a table of values.
Vertical asymptote: x = 0

16. Example 4B: Graphing Rational Functions with Vertical and Horizontal Asymptotes
Identify the zeros and asymptotes of the function. Then graph.
x – 2
x2 – 1
f(x) =
x – 2
(x – 1)(x + 1)
f(x) =
Factor the denominator.
The numerator is 0 when x = 2.
Zero: 2
Vertical asymptote: x = 1, x = –1
The denominator is 0 when x = ±1.
Horizontal asymptote: y = 0
Degree of p < degree of q.

17. Example 4B Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph with a graphing calculator or by using a table of values.

18. Identify the zeros and asymptotes of the function. Then graph.
Example 4C: Graphing Rational Functions with Vertical and Horizontal Asymptotes
Identify the zeros and asymptotes of the function. Then graph.
4x – 12
x – 1
f(x) =
4(x – 3)
x – 1
f(x) =
Factor the numerator.
The numerator is 0 when x = 3.
Zero: 3
Vertical asymptote: x = 1
The denominator is 0 when x = 1.
The horizontal asymptote is y = = = 4. 4 1
leading coefficient of p
leading coefficient of q
Horizontal asymptote: y = 4

19. Example 4C Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph with a graphing calculator or by using a table of values.

20. Check It Out! Example 4a
Identify the zeros and asymptotes of the function. Then graph.
x2 + 2x – 15
x – 1
f(x) =
(x – 3)(x + 5)
x – 1
f(x) =
Factor the numerator.
The numerator is 0 when x = 3 or x = –5.
Zeros: 3 and –5
The denominator is 0 when x = 1.
Vertical asymptote: x = 1
Horizontal asymptote: none
Degree of p > degree of q.

21. Check It Out! Example 4a Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph with a graphing calculator or by using a table of values.

22. Check It Out! Example 4b
Identify the zeros and asymptotes of the function. Then graph.
x – 2
x2 + x
f(x) =
x – 2
x(x + 1)
f(x) =
Factor the denominator.
The numerator is 0 when x = 2.
Zero: 2
Vertical asymptote: x = –1, x = 0
The denominator is 0 when x = –1 or x = 0.
Horizontal asymptote: y = 0
Degree of p < degree of q.

23. Check It Out! Example 4b Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph with a graphing calculator or by using a table of values.

24. Identify the zeros and asymptotes of the function. Then graph.
Check It Out! Example 4c
Identify the zeros and asymptotes of the function. Then graph.
3x2 + x
x2 – 9
f(x) =
x(3x – 1)
(x – 3) (x + 3)
f(x) =
Factor the numerator and the denominator.
Zeros: 0 and – 1 3
The numerator is 0 when x = 0 or x = – . 1 3
Vertical asymptote: x = –3, x = 3
The denominator is 0 when x = ±3.
The horizontal asymptote is y = = = 3. 3 1
leading coefficient of p
leading coefficient of q
Horizontal asymptote: y = 3

25. Check It Out! Example 4c Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph with a graphing calculator or by using a table of values.