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

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

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

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.

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.

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

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

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

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