Chapter 4: Polynomial & Rational Functions 4.4: Rational Functions

1. Chapter 4: Polynomial & Rational Functions4.4: Rational Functions Essential Question: How can you determine the vertical and horizontal asymptotes of an equation?

2. 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

3. 4.4: Rational Functions • Properties of Rational Graphs • 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. 4.4: 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.

5. 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

6. 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

7. 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.

8. 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

9. 4.4: Rational Functions • Asymptotes, Example #2 • Vertical asymptotes at: • x2 – 4 = 0 • (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.

10. 4.4: Rational Functions • Asymptotes, Example #3 • Vertical asymptotes at: • x3 + 1 = 0 • Graphing tells you there is only one root, at x = -1 • Vertical asymptotes at x = -1 • Horizontal Asymptotes at: • Horizontal asymptote at y = 2.

11. 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)