1. 2.7 Rational Functions By: Meteor, Al Caul, O.C., and The Pizz

2. Rational functions A rational function can be written in the form f(x)= p(x) q(x) p(x) and q(x) are polynomials q(x) is not the zero polynomial p(x) and q(x) has no common factors the domain of s rational function of x includes all real numbers except x-values that makes the denominator

3. Example 1 finding the domain of a rational function Find the domain of f(x)= 1/x Denominator is zero when x=0, so it can’t be 0 Domain is all real numbers execpt 0 So start with -1 x -1 -0.5 -0.1 -0.01 -0.001 0 f(x) -1 -2 -10 -100 -1000 -∞

4. Asymptotes of a Rational Function f(x)= 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 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has one or no horizontal asymptote determined by the following rules a. If n < m, the graph of f has the x-axis or ( y=0 ) as a horizontal asymptote b. If n = m, the graph of f has the line y = a n / b m as a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote

5. Finding the Asymptotes f(x) = 2 + x 2 - x Remember!!! 1. To find the x-intercept get the top of the equation equal to 0 2. To find the y-intercept put 0 in for all x’s and solve 3. To find the vertical asymptote set the bottom equation to 0 and solve for it 4. To find the horizontal asymptote remember that the leading coefficient exponent on on top is N and the leading coefficient exponent on the bottom is M - N = M leading coefficients are horizontal asymptote - N > M no horizontal asymptote - N < M Horizontal asymptote is the x-axis or 0

6. Example #2 f(x) = 2 + x 2 - x - 1 st step find the x-intercept - get the top of the equation equal to 0 2 + x = 0 -Subtract the 2 X= -2 The x-intercept is -2

7. Example #2 f(x) = 2 + x 2 - x 2 nd Step Find the y-intercept - Put 0 in for all x’s and solve 2 + 0 2 – 0 Which all ends up equaling 1 The y-intercept is 1

8. Example #2 f(x) = 2 + x 2 - x 3 rd Step find the vertical asymptote - set the bottom equation to 0 and solve for it 2 – x = 0 You can subtract the 2 to get -x = -2 or x = 2 The vertical asymptote is at x = 2

9. Example #2 f(x) = 2 + x 2 - x 4 th step is find the horizontal asymptote - the leading coefficient exponent on on top is N and the leading coefficient exponent on the bottom is M - N = M leading coefficients are horizontal asymptote - N > M no horizontal asymptote - N < M Horizontal asymptote is the x-axis or 0 Since the leading coefficient exponents are equal the leading coefficients are horizontal asymptote 2 or 1 2 The horizontal asymptote is at y= 1

10. Grand Finale f(x) = 2 + x 2 - x x-intercept is -2 y-intercept is 1 vertical asymptote is at x = 2 horizontal asymptote is at y= 1

11. Sketching the Graph of Rational Fractions Guidelines: –Find and plot y-intercept. –Find the zeros of the numerator, then plot on x-axis. –Find the zeros of the denominator, then plot vertical asymptotes. –Find and sketch the horizontal asymptotes. –Plot at least one point between and beyond each x- intercept and vertical asymptotes.

12. Sketching Ex. 3 Let F(x)= p(x)/q(x) –F(x)=1-3x 1-x Y-int=1 X-int.=1/3 Vertical-x=1 Horizontal-y=3

13. Sketching Ex. 4 F(x)= x x 2 -x-2 X-int.=0 Y-int.=0 Vertical- x=-1,2 Horizontal- y=0

14. Slant Asymptotes Guidelines: –If the degree of the numerator of a rational function is exactly one more than the degree of the denominator, the graph of the function has a slant asymptote. –For Example: x 2 -x is a slant asymptote because it x+1 is exactly one more on the top than the bottom.

15. Slants To find a slant asymptote, use polynomial or synthetic division. Ex. 5 F(x)= x 3 x 2 -1 y-int=0 X-int=0 Vertical=+1 Horizontal=none Slant-y=x

16. Slants Ex. 6 F(x)=1-x = 1-x 2 x x Y-int=none X-int=+1 Vertical-x=0 Horizontal=none Slant-y=-x