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 (boundary lines). The f(x) = has a vertical asymptote at x = 0 and a horizontal asymptote at
y = 0. 1 x

2. Notes: Graphing Hyperbolas1
x +6
1A . Graph g(x) =
- 1
x +6
1B. Graph g(x) = 2 x
2 . Graph g(x) =
3. Identify the asymptotes, domain, and range of the function g(x) = – 4. 1
x +6

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

5. 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.

6. 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 (boundary lines). The f(x) = has a vertical asymptote at x = 0 and a horizontal asymptote at
y = 0. 1 x

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. Notes: Graphing Hyperbolas1
x +6
1A . Graph g(x) =
- 1
x +6
1B. Graph g(x) = 2 x
2 . Graph g(x) =

9. Notes: Graphing Hyperbolas
3. Identify the asymptotes, domain, and range of the function g(x) = – 4. 1
x +6
Vertical asymptote: x = –6
Domain: all reals except x ≠ –6
Horizontal asymptote: y = –4
Range: all reals except y ≠ –4