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Warm up

Warm up. Describe the end behavior of: Graph the function Determine the interval(s) on which the function is increasing and on which it is decreasing. . Lesson 3-7 Graphs of Rational Functions. Objective: 1. To graph rational functions.

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Warm up

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  1. Warm up • Describe the end behavior of: • Graph the function Determine the interval(s) on which the function is increasing and on which it is decreasing.

  2. Lesson 3-7 Graphs of Rational Functions Objective: 1. To graph rational functions. 2. To determine vertical, horizontal and slant asymptotes

  3. Rational Functions • Rational functions are the quotient of 2 polynomial functions where h(x) ≠ 0. • The parent function is

  4. Asymptotes • Asymptotes are lines that a graph approaches, but does not intersect. • We will look at vertical, horizontal and slant asymptotes.

  5. Asymptotes • A rational function can have more than one vertical asymptote, but it usually has one horizontal asymptote at most.

  6. Vertical Asymptotes Rational functions are the quotient of 2 polynomial functions where h(x) ≠ 0. If g(x) and h(x) have no common factors, then f(x) has vertical asymptote(s) when h(x) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator. (where the denom. is undefined.)

  7. Vertical Asymptotes x=a is a vertical asymptote for f(x) if or as from either the left or the right.

  8. Example • For , the vertical asymptote is x = 0

  9. Example • Find the vertical asymptote of • Since the function is undefined at 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1.

  10. Horizontal Asymptote • To determine or prove the horizontal asymptote: • Find the highest degree variable in the denominator. • Divide each term in the function by this.

  11. Horizontal Asymptote highest degree variable is x

  12. example cont’d is the horizontal asymptote

  13. SHORT CUTS If the degree of g(x) is less than the degree of h(x), then the horizontal asymptote isy = 0. (this could be y= some vertical shift also) b. If the degree of g(x) is equal to the degree of h(x), then the horizontal asymptote is c. If the degree of g(x) is greater than the degree of h(x), then there is no horizontal asymptote.

  14. Horizontal Asymptotes Degree of numerator = 1 Degree of denominator = 2 Example: Find the horizontal asymptote: Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote is y = 0.

  15. Horizontal Asymptotes Degree of numerator = 1 Degree of denominator = 1 Example: Find the horizontal asymptote: Since the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is .

  16. Horizontal Asymptotes Example: Degree of numerator = 2 Degree of denominator = 1 Find the horizontal asymptote: Since the degree of the numerator is greater than the degree of the denominator, there is nohorizontal asymptote.

  17. exceptions • If there are 2 vertical asymptotes, the horizontal asymptote may or may not hold. • vert. asymp at x=.562 and -3.562 • horiz. aymp at y = 1

  18. exceptions

  19. Vertical & Horizontal Asymptotes Practice: Find the vertical and horizontal asymptotes:

  20. Slant asymptotes • Slant asymptotes occur when the degree of the numerator of a rational function is 1 more than the degree of the denominator. Find by using polynomial long division. • ex:

  21. Slant asymptotes • Using polynomial long division will yield • As , therefore the slant asymptote is y = x.

  22. Practice • Find the slant asymptote for

  23. Exceptions • Some rational functions will only have point discontinuity instead of an asymptote. • This occurs whenever the numerator and the denominator share a common factor.

  24. Exceptions • Find the asymptotes for No vertical asymptote No horizontal asymptote Because numer. & denom. have (x-2) in common there is also no slant asymptote – but a hole(point of discontinuity) at x=2.

  25. sources teachers.henrico.k12.va.us/math/hcpsalgebra2/.../AII7_E_asymptotes.ppt, Sept 17,2013 WolframAlpha. Wolfram Alpha LLC, 2013. Web. 20 Sept. 2013. <http://www.wolframalpha.com>.

  26. Find the horizontal asymptote: Exponents are the same; divide the coefficients Bigger on Top; None Bigger on Bottom; y=0

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