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

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

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) = + 2. 5 x – 1 3. Identify the zeros and asymptotes of the function. Then graph. 4x – 12 x – 1 f(x) =

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.

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

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}

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.

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

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.

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

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) = + 2. 5 x – 1 asymptotes: x = 1, y = 2; D:{x|x ≠ 1}; R:{y|y ≠ 2}

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

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