1. PreCalculus 2.6 – Rational Functions

2. Example 1 – Finding the Domain of a Rational Function Find the domain of f(x) = 1/x and discuss the behavior of f near any excluded x-values.

4. Horizontal and Vertical Asymptotes • Asymptotes – A line that the graph of a function approaches more and more closely without ever touching.

5. Asymptotes of a Rational Function • The graph of f has vertical asymptotes at the zeros of the denominator of the function. • The graph of f has one or no horizontal asymptote determined by comparing the degrees of the numerator and denominator. • If the degree of the numerator is less than the denominator, the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote.

6. If the degree of the numerator is equal to the denominator, the graph of f has the line resulting in dividing the leading term of the numerator by the leading term of the denominator as the horizontal asymptote. • If the degree of the numerator is greater than the degree of the denominator, the graph of f has no horizontal asymptote.

7. Example 2 – Finding Horizontal and Vertical Asymptotes • Find all vertical and horizontal asymptotes of the graph of each rational function. A) f (x) = B) f (x) =

9. Example 3 – Finding Horizontal and Vertical Asymptotes • Find all vertical and horizontal asymptotes of the graph of each rational function. A) f (x) = B) f (x) =

11. Analyzing Graphs of Rational Functions • Let f(x) = g(x) / h(x) , where g(x) and h(x) are polynomials with no common factors. • Find and plot the y-intercept (if any) by evaluating f(0) • Find the zeros of the numerator (if any) by solving the equation g(x) = 0. Then plot the corresponding x-intercepts. • Find the zeros of the denominator (if any) by solving the equation h(x) = 0. Then sketch the corresponding vertical asymptotes.

12. Find and sketch the horizontal asymptote (if any) by using the rule for finding the horizontal asymptote of a rational function. • Test for symmetry • Plot at least one point between and one point beyond each x-intercept and vertical asymptote. • Use smooth curves to complete the graph between and beyond the vertical asymptotes.

13. Example 4 – Sketching the Graph of a Rational Function Sketch the graph of g(x) = and state its domain.

14. Example 5 – Sketching the Graph of a Rational Function Sketch the graph of f(x) =

15. Example 6 – Sketching the Graph of a Rational Function Sketch the graph of f(x) =

16. Example 7 – Sketching the Graph of a Rational Function Sketch the graph of f(x) =

17. Slant Asymptotes • If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. • To find the equation of a slant asymptote, use long division.

18. Example 8 – A Rational Function with a Slant Asymptote Find any slant asymptotes for each of the rational functions below. A) f (x) = B) f (x) =