1. Key Information Starting Last Unit Today –Graphing –Factoring –Solving Equations –Common Denominators –Domain and Range (Interval Notation) Factoring will be critical next week Retest is this Thursday PM and Friday AM Must have test corrections done by then Factoring Quiz Tuesday. Only 2 left!!

2. Warm-up

3. Section 8-2 & 8-3 Graphing Rational Functions

4. Objectives I can determine vertical asymptotes of a rational function and graph them I can determine horizontal asymptotes of a rational function and graph them I can find x and y intercepts to help graph I can graph rational functions using a calculator

5. Rational Functions A rational function is any ratio of two polynomials, where denominator cannot be ZERO! Examples:

6. Asymptotes Asymptotes are the boundary lines that a rational function approaches, but never crosses. We draw these as Dashed Lines on our graphs. There are two types of asymptotes we will study in Alg-2: –Vertical –Horizontal

7. Vertical Asymptotes Vertical Asymptotes exist where the denominator would be zero. They are graphed as Vertical Dashed Lines There can be more than one! To find them, set the denominator equal to zero and solve for “x”

8. Example #1 Find the vertical asymptotes for the following function: Set the denominator equal to zero x – 1 = 0, so x = 1 This graph has a vertical asymptote at x = 1

9. 1263457891010 4 3 2 7 5 6 8 9 x- axis y- axis 0 1-2-6 -3-4-5 -7-8-9 1010 -4 -3 -2 -7 -5 -6 -8 -9 0 Vertical Asymptote at X = 1

10. Other Examples: Find the vertical asymptotes for the following functions:

11. Horizontal Asymptotes Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary. To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator See next slide:

12. Horizontal Asymptote Given the Rational Function: Compare DEGREE of Numerator to Denominator If N < D, then y = 0 is the HA If N = D, then the HA is If N > D, then the graph has NO HA

13. Example #1 Find the horizontal asymptote for the following function: Since the degree of numerator is equal to degree of denominator (m = n) Then HA: y = 1/1 = 1 This graph has a horizontal asymptote at y = 1

14. 1263457891010 4 3 2 7 5 6 8 9 x- axis y- axis 0 1-2-6 -3-4-5 -7-8-9 1010 -4 -3 -2 -7 -5 -6 -8 -9 0 Horizontal Asymptote at y = 1

15. Other Examples: Find the horizontal asymptote for the following functions:

16. Intercepts x-intercepts (there can be more than one) Set Numerator = 0 and solve for “x” y-intercept (at most ONE y-intercept) Let all x’s =0 and solve

17. Graphing Rational Expressions Factor rational expression and reduce Find VA (Denominator = 0) Find HA (Compare degrees) Find x-intercept(s) (Numerator = 0) Find y-intercept (All x’s = 0) Next type the function into the graphing calculator and look up ordered pairs from the data table to graph the function. Remember that the graph will never cross the VA

18. Calculator

19. Key Data to Graph

20. Graph: f(x) = Vertical asymptote: x – 2 = 0 so at x = 2 Dashed line at x = 2 m = 0, n = 1 so m<n HA at y = 0 No x-int y-int = (0, -1) Put into graphing calc. Pick ordered pairs

21. f(x) = Vertical Asymptotes: x – 2 = 0 and x + 3 = 0 x = 2, x = -3 m = 0, n = 2 m < n HA at y = 0 No x-int y-int (0, -1) Graph on right

22. Calculator

23. Homework WS 12-1 Must know how to factor for next week!!! Factoring Quiz Tuesday