1. ACTIVITY 35: Rational Functions (Section 4.5, pp. 356-369)

2. Rational Functions and Asymptotes: A rational function is a function of the form where P(x) and Q(x) are polynomials (that we assume without common factors). When graphing a rational function we must pay special attention to the behavior of the graph near the x-values which make the denominator equal to zero.

3. Example 1: Find the domain of each of the following rational functions: The domain well be all x’s such that x 2 – 4 ≠ 0. Consequently, the domain is all x’s except -2 and 2. That is, The domain well be all x’s such that x 2 + 4 ≠ 0. This is impossible! Consequently, the domain is all real numbers.

4. The domain well be all x’s such that x 2 – 4x ≠ 0. Consequently, the domain is all x’s except 0 and 4. That is,

5. Example 2: Sketch the graphs of: xy 0 undefined 11 21/2 31/3 1/22 1/33 1/44 xy -2-1/2 -3-1/3 -1/2-2 -1/3-3 -1/4 -4

6. Hint: observe that g(x) = f(x − 2) + 3

7. Definition of Vertical and Horizontal Asymptotes: The line x = a is a vertical asymptote of y = r(x) if y → ∞ or y → −∞ as x → a + or x → a − Note: x → a − means that “x approaches a from the left”; x → a + means that “x approaches a from the right”.

8. The line y = b is a horizontal asymptote of y = r(x) if y → b as x → ∞ or x → −∞

9. Asymptotes of Rational Functions: The vertical asymptotes of r(x) are the lines x = a, where a is a zero of the denominator. If n < m, then r(x) has horizontal asymptote y = 0. If n = m, then r(x) has horizontal asymptote y =. If n > m, then r(x) has no horizontal asymptote. Let r(x) be the rational function:

10. Example 3: Find the horizontal and vertical asymptotes of each of the following functions: Vertical Asymptotes Horizontal Asymptotes Consequently, r(x) has horizontal asymptote y = 0.

11. Vertical Asymptotes Horizontal Asymptotes Consequently, r(x) has horizontal asymptote y =

12. Vertical Asymptotes Horizontal Asymptotes Consequently, r(x) has no horizontal asymptote.

13. Example 4: Graph the rational function This is impossible so there are NO x-intercepts So the y-intercept is (0,-1)

14. Vertical Asymptotes Horizontal Asymptotes Consequently, r(x) has horizontal asymptote y = 0.

15. 1.No x-intercepts 2.y-intercept is (0,-1) 3.Vertical Asymptotes x = – 1 and x = 6 4.Horizontal Asymptotes y = 0

16. Example 5: Graph the rational function So the y-intercept is

17. Vertical Asymptotes Horizontal Asymptotes Consequently, r(x) has horizontal asymptote y =.

18. 1.x-intercepts (1,0) and (-2,0) 2.y-intercept is (0,2/3) 3.Vertical Asymptotes x = – 1 and x = 3 4.Horizontal Asymptotes y = 1