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. 1 x
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

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. Notes: 1 x
1. Using the graph of f(x) = as a guide, describe the transformation and graph the function g(x) = 1
x – 4
2.Identify the asymptotes, domain, and range of the function g(x) = 5
x – 1
3. Identify the zeros and asymptotes of the function. Then graph.
4x – 12
x – 1
f(x) =

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

5. 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) =
translate f 2 units left.
translate f 3 units down.

6. Example 2
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) =
translate f 1 unit up.
translate f 4 units left.

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

8. Example 3: Determining Properties of Hyperbolas
Identify the asymptotes, domain, and range of the function g(x) = – 2. 1
x + 3
Vertical asymptote: x = –3
Domain: {x|x ≠ –3}
Horizontal asymptote: y = –2
Range: {y|y ≠ –2}

11. 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.

12. Identify the zeros and asymptotes of the function. Then graph.
Example 4A Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph using a table of values.
Vertical asymptote: x = 0

13. 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.

14. Example 4B Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph using a table of values.

15. Notes: 1 x
1. Using the graph of f(x) = as a guide, describe the transformation and graph the function g(x) = 1
x – 4
g is f translated 4 units right. 2.
Identify the asymptotes, domain, and range of the function g(x) = 5
x – 1
asymptotes: x = 1, y = 2; D:{x|x ≠ 1}; R:{y|y ≠ 2}

16. 3. Identify the zeros and asymptotes of the function. Then graph.
Notes #3:
3. 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

17. Notes #3 Continued
Identify the zeros and asymptotes of the function. Then graph.
Graph using a table of values.