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Section 2.7

Section 2.7. By Joe, Alex, Jessica, and Tommy. Introduction. Any function can be written however you want it to be written A rational function can be written this form: f(x)=p(x)/q(x) p(x) and q(x) are polynomials q(x) is not equal to zero. Asymptotes.

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Section 2.7

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  1. Section 2.7 By Joe, Alex, Jessica, and Tommy

  2. Introduction • Any function can be written however you want it to be written • A rational function can be written this form: • f(x)=p(x)/q(x) • p(x) and q(x) are polynomials • q(x) is not equal to zero

  3. Asymptotes Asymptotes are the lines that a rational function cannot cross Three kinds: Vertical, Horizontal, and Slant Vertical asymptote occurs when q(x) (the bottom of the function) equals zero In graph shown, Vert. Asym. is zero

  4. Asymptotes contd. p(x)=anxn+an-1xn-1+…a1x+a0 q(x) = bmxm+bm-1xm-1+…b1x+b0 • Horizontal asymptotes are found one of three ways • Y=0 if n<m • None if n>m Look for slant asymptote (see another slide) • an/bm if n=m divide leading coefficients iff’n the power is the same

  5. Super Slant Asymptotes • If n>m and n is exactly one bigger than m, use synthetic division to find a slant Asymptote. • Discard remainder and the slant is the quotient • x2-x-2/x-1=x remainder is -2 • Slant is y=x y=x f(0)=2 (-1,0) (2,0) x=1

  6. Sketching Rational Functions • Find and plot the y-intercept (if any) by evaluating f(0) • Find and sketch any x-intercepts (if any) by evaluating p(x)=0. • Find and sketch any vertical asymptotes (if any) by evaluating q(x)=0 • Find and sketch the horizontal asymptotes (if any) by using the rules for finding a horizontal asymptote (see slide 4) • Plot at least one point betweenand at leastone point beyond each x-intercept and vertical asymptote • Use smooth curves to complete the graph between and beyond the vertical asymptotes

  7. a) 2x 3x2+1 b) 2x2 3x2+1 c) 2x3 3x2+1 a)The x-axis is the horizontal asymptote because the degree of the numerator is less than the degree of the denominator b) The line y=2/3 is the horizontal asymptote because the degrees of the numerator and denominator are the same, so an/bm=2/3 c) There is no horizontal asymptote, but since the numerator is exactly one degree bigger than the denominator you can use synthetic division to find the slant asymptote Example 1:Finding Horizontal Asymptotes

  8. Sketch: g(x)= 3 x-2 y-intercept: (0, -3/2) x-intercept: none because 3 = 0 Vertical asymptote: x=2 Horizontal asymptote: y=0 degree of p(x)<degree of q(x) Other points: Example 2: Sketching Rational Functions

  9. The end

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