Chapter 4: Polynomial & Rational Functions 4.4: Rational Functions Essential Question: How can you determine the vertical and horizontal asymptotes of an equation?

4.4: Rational Functions Domain of Rational Functions The domain is the set of all real numbers that are not zeros of its denominator. Example 1: The Domain of a Rational Function Find the domain of each rational function All real numbers except x = 0 All real numbers except x2 – x – 6 = 0 (x – 3)(x + 2) = 0, so all real numbers except for 3 and -2

4.4: Rational Functions Properties of Rational Graphs Example Intercepts As with any graph, the y-intercept is at f(0) The x-intercepts are when the numerator = 0 and the denominator does not equal 0. Example Find the intercepts of y-intercept: x-intercepts: Neither are solutions of x – 1 = 0 so both are x-intercepts

4.4: Rational Functions Properties of Rational Graphs (continued) Vertical Asymptotes Whenever only the denominator = 0 (numerator is not 0) Vertical asymptotes will either spike up to ∞ or down to -∞ Big-Little Concept Dividing by a small number results in a large number Dividing by a large number results in a small number.

4.4: Rational Functions Behavior near a Vertical Asymptote Describe the graph of near x = 2 As x gets closer and closer to 2 from the right (2.1, 2.01, 2.001, …) the denominator becomes a really small positive number. Division by a small positive number means the graph of f(x) approaches ∞ from the right As x gets closer and closer to 2 from the left (1.9, 1.99, 1.999, …) the denominator becomes a really small negative number Division by a small negative number means the graph of f(x) approaches -∞ from the left

4.4: Rational Functions Holes When a number c is a zero of both the numerator and denominator of a rational function, the function might have a vertical asymptote, or it might have a hole. Example #1 But this is not the same as the function g(x) = x + 2, as f(2) = while g(2) = 4, so though they may look the same, f(x) has a hole at x = 2 Example #2 The graph of x2/x3 looks the same as 1/x, and has a vertical asymptote, as neither of the functions are defined at x = 0

4.4: Rational Functions Holes If and a number d exists such that g(d) and h(d) = 0 If the degree of the numerator is greater than (or equal to) the degree of the denominator after simplification, then the function has a hole at x = d = hole @ x = 5 If the degree of the denominator is greater than the degree of the numerator after simplification, then the function has a vertical asymptote at x = d = asymptote @ x = 5 It is far easier to first determine the domain of the function, and then visually inspect to see whether “hiccups” in the domain are holes or asymptotes.

4.4: Rational Functions End Behavior (Horizontal Asymptotes) The horizontal asymptote is found by determining what the function will be when x is extraordinarily large When x is large, a polynomial function behaves like its highest degree term Example #1 List the vertical asymptotes and describe the end behavior. There is a vertical asymptote at x = 5/2 Both numerator and denominator have the same degree, so the horizontal asymptote is at y = -3/2

4.4: Rational Functions Asymptotes, Example #2 Vertical asymptotes at: (x – 2)(x + 2) = 0 Vertical asymptotes at x = 2 or x = -2 Horizontal Asymptotes at: As x becomes large, 1/x becomes small. So horizontal asymptote at y = 0.

4.4: Rational Functions Asymptotes, Example #3 Vertical asymptotes at: Graphing tells you there is only one root, at x = -1 Vertical asymptotes at x = -1 Horizontal Asymptotes at: Horizontal asymptote at y = 2.

4.4: Rational Functions Assignment Page 290 1 – 49, odd problems Due tomorrow Show work Only worry about holes, vertical and horizontal asymptotes. Disregard stuff on slant/parabolic asymptotes Ignore the graphing (you have a graphing calculator for that)