1. 4.5 Rational Functions  For a rational function, find the domain and graph the function, identifying all of the asymptotes.

2. Rational Function A rational function is a function f that is a quotient of two polynomials, that is, where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x)  0.

3. Example Determine the domain of each of the functions shown below.

4. Example Determine the Domain of the following.

5. The line x = k is a vertical asymptote of the graph of f if f(x)  ∞ or f(x)  –∞ as x approaches k from either the left or the right. Vertical Asymptote x = 2

6. Let f be a rational function given by written in lowest terms. To find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote. Example: Finding Vertical Asymptotes

7. Find the vertical asymptotes of the function and then graph the function on your graphing calculator. a) b) c)

8. The line y = k is a horizontal asymptote of the graph of f if f(x)  k as x  ∞ or f(x)  k as x  –∞. Horizontal Asymptotes

9. Horizontal Asymptote (a)If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote. (b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator. (c)If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes. Finding Horizontal Asymptotes

10. Finding Oblique Asymptotes (d)An oblique asymptote occurs when the degree of the numerator is 1 greater than the degree of the denominator. There can be only one horizontal asymptote or one oblique asymptote and never both. REMINDER: An asymptote is not part of the graph of the function.

11. Find the horizontal/oblique asymptote of the functions below. a) b) c) d)

12. Oblique or Slant Asymptote Find all the asymptotes of. Divide to find an equivalent expression. The line y = 2x  1 is an oblique asymptote.

13. To determine whether the graph will intersect its horizontal/slant asymptote at y = k, set the f(x) = k and solve. If there is no solution the graph will not cross the asymptote. Function Crosses Horizontal/Slant Asymptote?

14. Determine algebraically if the graph of the function will cross its horizontal/slant asymptote. If so, where?

15. Holes In The Graphs If f(x) = p(x)/q(x), then it is possible that, for some number k, both p(k) = 0 and q(k) = 0. In this case, the graph of f may not have a vertical asymptote at x=k; rather it may have a “hole” at x=k.

16. “Holes in a Graph” Find any holes in graph.

17. To graph a rational function, f (x)=p(x)/q(x) 1. Determine the domain of the function and restrict any x- values as needed. Holes in Graph? 2. Find and plot the y-intercept (evaluate f (0)). 3. Find and plot any x-intercepts (solve p(x)=0). 4. Find any vertical asymptotes (solve q(x)=0), if there is any. 5. Find the horizontal/slant asymptote, if there is one. Determine whether the graph will cross its horizontal/slant asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal/Slant asymptote. 6. Plot at least one point between x-intercepts and vertical asymptotes to determine the behavior of the graph. 7. Complete the sketch. Sketching the Graph of a Rational Function

18. Graph Hole located at (0, 4) 1.Hole in Graph? If so, where? 2.Vertical Asymptote(s) 3.Horizontal/Oblique Asymptote 4. Cross? If so, where? 5.x-intercept(s) 6. y-intercept Yes, (0, 4) x =  1, x = 1 y = 0 Yes, (4, 0) (4, 0) None

19. Graph 1.Hole in Graph? If so, where? 2.Vertical Asymptote(s) 3.Horizontal/Oblique Asymptote 4. Cross? If so, where? 5.x-intercept(s) 6. y-intercept No Hole in Graph x =  2 y = 1 No crossing (3, 0) (0,  3/2)

20. Graph 1. Hole in Graph? If so, where? 2. Vertical Asymptote(s) 3. Horizontal/Oblique Asymptote 4. Cross? If so, where? 5. x-intercept(s) 6. y-intercept No Hole in Graph x =  1 y = x  1 No crossing (0, 0)