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Understanding Rational Functions and Asymptotes

Learn about rational functions, domain, vertical and horizontal asymptotes, and graphing techniques. Explore examples and understand the behavior of graphs near x-values making the denominator zero.

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Understanding Rational Functions and Asymptotes

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  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 x2 – 4 ≠ 0. Consequently, the domain is all x’s except -2 and 2. That is, The domain well be all x’s such that x2 + 4 ≠ 0. This is impossible! Consequently, the domain is all real numbers.

  4. The domain well be all x’s such that x2 – 4x ≠ 0. Consequently, the domain is all x’s except 0 and 4. That is,

  5. Example 2: Sketch the graphs of: -1 0 undefined -1 1 -2 1 -1/2 2 -3 1/2 -1/3 -1/2 3 1/3 -2 1/2 -1/3 2 -3 -1/4 1/3 3 -4 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: Let r(x) be the rational function: 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.

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

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

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

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

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

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

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

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

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

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