Rational Functions and Models 4.6 Identify a rational function and state its domain Identify asymptotes Interpret asymptotes Graph a rational function by using transformations Graph a rational function by hand Copyright © 2010 Pearson Education, Inc.

Rational Function A function f represented by where p(x) and q(x) are polynomials and q(x) ≠ 0, is a rational function. Copyright © 2010 Pearson Education, Inc.

Rational Function The domain of a rational function includes all real numbers except the zeros of the denominator q(x). The graph of a rational function is continuous except at x-values where q(x) = 0. Copyright © 2010 Pearson Education, Inc.

Example 1 For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Copyright © 2010 Pearson Education, Inc.

Example 1 Solution a) b) Is a rational function - both numerator and denominator are polynomials; domain is all real numbers; x2 + 1 ≠ 0 Is NOT a rational function Denominator is not a polynomial; domain is {x | x > 0} Copyright © 2010 Pearson Education, Inc.

Example 1 Solution continued c) Is a rational function - both numerator and denominator are polynomials; domain is {x | x ≠1, x ≠ 2} because (x – 1)(x – 2) = 0 when x = 1 and x = 2. Copyright © 2010 Pearson Education, Inc.

Vertical Asymptotes The line x = k is a vertical asymptote of the graph of f if f(x) g ∞ or f(x) g –∞ as x approaches k from either the left or the right. Copyright © 2010 Pearson Education, Inc.

Horizontal Asymptotes The line y = b is a horizontal asymptote of the graph of f if f(x) g b as x approaches either ∞ or –∞. Copyright © 2010 Pearson Education, Inc.

Finding Vertical & Horizontal Asymptotes Let f be a rational function given by written in lowest terms. Vertical Asymptote To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote. Caution: If k is a zero of both q(x) and p(x), then f(x) is not written in lowest terms, and x – k is a common factor. Copyright © 2010 Pearson Education, Inc.

Finding Vertical & Horizontal Asymptotes (a) If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote. (b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. Copyright © 2010 Pearson Education, Inc.

Finding Vertical & Horizontal Asymptotes (c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes. Copyright © 2010 Pearson Education, Inc.

Example 4 For each rational function, determine any horizontal or vertical asymptotes. Solution g(x) is a translation of f(x) left one unit and down 2 units. The vertical asymptote is x = 1 The horizontal asymptote is y = 2 g(x) = f(x + 1)  2 Copyright © 2010 Pearson Education, Inc.

Example 4 For each rational function, determine any horizontal or vertical asymptotes. a) b) c) Copyright © 2010 Pearson Education, Inc.

Example Solution a) Degree of numerator and denominator are both 1. Since the ratio of the leading coefficients is 6/3, the horizontal asymptote is y = 2. When x = –1, the denominator, 3x + 3, equals 0 and the numerator, 6x – 1 does not equal 0, so the vertical asymptote is x = 1 Copyright © 2010 Pearson Education, Inc.

Example Solution continued a) Here’s a graph of f(x). Copyright © 2010 Pearson Education, Inc.

Example Solution continued b) Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote. When x = ±2, the denominator, x2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes are x = 2 and x = 2. Copyright © 2010 Pearson Education, Inc.

Example Solution continued b) Here’s a graph of g(x). Copyright © 2010 Pearson Education, Inc.

Example Solution c) Degree of numerator is greater than the degree of the denominator so there are no horizontal asymptotes. When x = –1, both the numerator and denominator equal 0 so the expression is not in lowest terms: g(x) = x – 1, x ≠ –1. There are no vertical asymptotes. Copyright © 2010 Pearson Education, Inc.

Example Solution c) Here’s the graph of h(x). A straight line with the point (–1, –2) missing. Copyright © 2010 Pearson Education, Inc.

Slant, or Oblique, Asymptotes A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator. Copyright © 2010 Pearson Education, Inc.

Slant, or Oblique, Asymptotes The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f. Copyright © 2010 Pearson Education, Inc.

Graphs and Transformations of Rational Functions Graphs of rational functions can vary greatly in complexity. We begin by graphing and then use transformations to graph other rational functions. Copyright © 2010 Pearson Education, Inc.

Example 5 Sketch a graph of and identify any asymptotes. Solution Vertical asymptote: x = 0 Horizontal asymptote: y = 0 Copyright © 2010 Pearson Education, Inc.

Example 6 Use the graph of to sketch a graph of Include all asymptotes in your graph. Write g(x) in terms of f(x). Copyright © 2010 Pearson Education, Inc.

Example 6 Solution g(x) is a translation of f(x) left 2 units and then a reflection across the x-axis. Vertical asymptote: x = –2 Horizontal asymptote: y = 0 g(x) = –f(x + 2) Copyright © 2010 Pearson Education, Inc.

Example 7 Let a) Use a calculator to graph f. Find the domain of f. b) Identify any vertical or horizontal asymptotes. c) Sketch a graph of f that includes the asymptotes. Copyright © 2010 Pearson Education, Inc.

Example 7 Solution a) Here’s the calculator display using “Dot Mode.” The function is undefined when x2 – 4 = 0, or when x = ±2. The domain of f is D = {x|x ≠ 2, x ≠ –2}. Copyright © 2010 Pearson Education, Inc.

Example 7 Solution b) When x = ±2, the denominator x2 – 4 = 0 (the numerator does not), so the vertical asymptotes are x = ±2. Degree of numerator = degree of denominator, ratio of leading coefficients is 2/1 = 2, so the horizontal asymptote is y = 2. Copyright © 2010 Pearson Education, Inc.

Example 7 Solution c) Here’s another version of the graph. Copyright © 2010 Pearson Education, Inc.

Graphing Rational Functions by Hand Let define a rational function in lowest terms. To sketch its graph, follow these steps. STEP 1: Find all vertical asymptotes. STEP 2: Find all horizontal or oblique asymptotes. STEP 3: Find the y-intercept, if possible, by evaluating f(0). Copyright © 2010 Pearson Education, Inc.

Graphing Rational Functions by Hand STEP 4: Find the x-intercepts, if any, by solving f(x) = 0. (These will be the zeros of the numerator p(x).) STEP 5: Determine whether the graph will intersect its nonvertical asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal asymptote, or by solving f(x) = mx + b, where y = mx + b is the equation of the oblique asymptote. Copyright © 2010 Pearson Education, Inc.

Graphing Rational Functions by Hand STEP 6: Plot selected points as necessary. Choose an x-value in each interval of the domain determined by the vertical asymptotes and x-intercepts. STEP 7: Complete the sketch. Copyright © 2010 Pearson Education, Inc.

Example 8 Graph Solution STEP 1: Vertical asymptote: x = 3 STEP 2: Horizontal asymptote: y = 2 STEP 3: f(0) = , y-intercept is Copyright © 2010 Pearson Education, Inc.

Example 8 Solution continued STEP 4: Solve f(x) = 0 The x-intercept is Copyright © 2010 Pearson Education, Inc.

Example 8 Solution continued STEP 5: Graph does not intersect its horizontal asymptote, since f(x) = 2 has no solution. STEP 6: The points are on the graph. STEP 7: Complete the sketch (next slide) Copyright © 2010 Pearson Education, Inc.

Example 8 Solution continued STEP 7 Copyright © 2010 Pearson Education, Inc.