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Published byRandolph Tracy Richard Modified over 7 years ago

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Add Holes

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Section 2.6 Rational Functions Grab out a calc!

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Exploration Graph using a calculator – what patterns do you notice?

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Rational Function Where N(x) and D(x) are polynomials and D(x) is not 0

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Find the domain of f(x)

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Steps to graph rational functions 1.Intercepts a.y-intercept b.x-intercept(s) 2. Asymptotes a. Vertical (0’s in the denominator) (x = ?) b. Horizontal (case 1, 2, or 3?) (y = ?) 3. Test for symmetry 4. Plot points! 5.Use smooth curves to complete the graph

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Asymptotes Vertical at zeros of the denominator Horizontal N < D asymptote at y = 0 N = D asymptote at y = a N / a D N > D no Horizontal asymptote

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Ex 1: Graph 1a. y-intercept when x = 0 f(0) = undefined No y-intercept!

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Graph 1b. Find the zeros of the numerator (x-intercepts) 0 = x 2 – 4 x = {-2, 2} = (x + 2)(x – 2)

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Graph 2a. Find the zeros of the Denominator (vertical asymptotes) 0 = x 2 x = {0} x = 0 is a vertical asymptote

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Graph 2b. Find and sketch the horizontal asymptote Since the degree of the numerator and the denominator are the same the horizontal asymptote will be y = 1/1=1 y = 1

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Graph 3. Test for Symmetry Since f(x)= f(-x) we know this drawing will be symmetrical about the y-axis

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Graph 4. Plot more points f(1)= f(-1) = -3 f(3)= f(-3) = 5/9

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Graph 5. Use smooth curves to complete the graph.

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Ex 2:Find the domain – try on your own and graph of g(x) Horizontal Asymptote at y = 0 when degree of numerator less than degree of the denominator. Vertical Asymptote at the zero of the denominator. y-int (0, -1/2) No x-int

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Ex 3: Graph f(x) No x or y intercepts

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Ex 3: Graph f(x) No x or y intercepts

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Ex 4: Graph Factor Horizontal asymptotes Vertical Asymptotes No x-int y-int (0, -1/6)

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Ex 4: Graph

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Ex 5: So what happens if there isn’t a horizontal asymptote??? Graph Lets rewrite the function to get a better idea of what’s happening Approaches 0 as x approaches infinity Thus, we have a slant asymptote! Slant Asymptote

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Ex 5: So what happens if there isn’t a horizontal asymptote??? Slant Asymptote

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Ex 6: Graph Lets rewrite the function to get a better idea of what’s happening Approaches 0 as x approaches infinity Thus, we have a slant asymptote! Slant Asymptote

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Ex 6: Graph Slant Asymptote

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Ex 7 Graph

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No asymptote at x = 2, but there is a hole!!!

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H Dub 2-6 Page 193 #13-20all, 21-33ODD, 38, 41, 45, 53, 56, 59

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