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) = an xn + an-1 xn-1 …... a1 x + a0 Q (x) bm xm + bm-1 xm-1 …... b1 x + b0 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 = an / bm as a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote

5. 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 Finding the Asymptotes

6. f(x) = 2 + x 2 - x - 1st 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 Example #2

7. f(x) = 2 + x 2 - x 2nd 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 Example #2

8. f(x) = 2 + x 2 - x 3rd 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 Example #2

9. f(x) = 2 + x 2 - x 4th 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 Example #2

10. 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 Grand Finale

11. 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. Sketching the Graph of Rational Fractions

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 x2-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: x2-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)= x3 x2-1 • y-int=0 • X-int=0 • Vertical=+1 • Horizontal=none • Slant-y=x

16. Slants Ex. 6 • F(x)=1-x = 1-x2 x x • Y-int=none • X-int=+1 • Vertical-x=0 • Horizontal=none • Slant-y=-x