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 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

Asymptotes contd. p(x)=a n x n +a n-1 x n-1 +…a 1 x+a 0 q(x) = b m x m +b m-1 x m-1 +…b 1 x+b 0 Horizontal asymptotes are found one of three ways Y=0 if n<m None if n>m Look for slant asymptote (see another slide) a n / b m if n=m divide leading coefficients iff’n the power is the same

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 x 2 -x-2/x-1=x remainder is -2 Slant is y=x f(0)=2 x 2 -x-2 (x+1)(x-2) (-1,0) (2,0) y=x x=1

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 between and at least one point beyond each x-intercept and vertical asymptote Use smooth curves to complete the graph between and beyond the vertical asymptotes

Example 1:Finding Horizontal Asymptotes a) 2x 3x 2 +1 b) 2x 2 3x 2 +1 c) 2x 3 3x 2 +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 a n /b m =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 2: Sketching Rational Functions 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: x-4135 g(x)

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