1. Introduction to Rational Functions Dr. Shildneck Fall, 2014

2. Definition A RATIONAL FUNCTION is a function in the form where p(x) and q(x) are polynomials and q(x) is not equal to zero.

3. Graphing Basic Rational Functions Type I: P(x) and Q(x) are linear The Parent Function is

4. The Graph is a HYPERBOLA The x-axis is a horizontal asymptote The y-axis is a vertical asymptote The domain is all real numbers except zero The range is all real numbers except zero The graph has two symmetrical parts, called branches. For each point (x, y) on one branch, there is a corresponding point (-x, -y) on the other branch.

5. The graph of

6. On Your Calculator… Graph

7. Describe what happens when… a > 1 in a < 0 in

8. All rational functions in the form have graphs that are called hyperbolas.

9. What Changes? Anything added to x shifts the graph left or right Anything added to the “whole” function shifts it up or down. Thus, the asymptotes are x = h, and y = k.

10. To Graph Draw the asymptotes Plot 2 points on each side of the vertical asymptote by choosing two x-values and plugging in. Use the points and the asymptotes as guides to draw the curve. The domain is all reals except h. The range is all reals except k.

11. Example 1 Graph:

12. All rational functions in the form also have hyperbolic graphs. Note that both the numerator and denominator are made up of linear functions.

13. The vertical asymptote will occur wherever the denominator is zero. The horizontal asymptote is y = a/c (where a and c are the leading coefficients) ASYMPTOTES of

14. To Graph Find and draw the asymptotes: y = a / c Where cx + d = 0 Choose x-values and plot 2 points on each side of the vertical asymptote. Draw the branches through the points, using the asymptotes as guidelines.

15. Example 2 Graph:

16. Real Life Uses The Junior class is sponsoring a dinner. The cost of catering the dinner is $9.95 per person plus a delivery charge of $18. (a)Write a model that gives the average cost per person. (b)Graph the model and use it to estimate the number of people needed to lower the cost to $11 per person. (c)Describe what happens to the average cost per person as the number of people increases.

17. Assignment Worksheet #1