Rational functions may have asymptotes (boundary lines). The f( x ) = has a vertical asymptote at x = 0 and a hor

1. 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 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

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

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. – 3 1 1 x + 2 x 1 x Example 1: Transforming Rational Functions Using the graph of f(x) = as a guide, describe the transformation and graph each function. A. g(x) = B. g(x) = translate f2 units left. translate f3 units down.

5. Using the graph of f(x) = as a guide, describe the transformation and graph each function. + 1 1 1 x + 4 x 1 x Example 2 a. g(x) = b. g(x) = translate f1 unit up. translate f4 units left.

6. 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 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

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. - 1 2 1 x +6 x +6 x Notes: Graphing Hyperbolas 1A . Graph g(x) = 1B. Graph g(x) = 2 . Graph g(x) = - 4

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