1. Graphing Rational Functions Objective: To graph rational functions without a calculator.

2. Points to find There are some specific things that we will find in order to graph a rational function.

3. Points to find There are some specific things that we will find in order to graph a rational function. 1.The y-intercept. Let x = 0 and solve for y.

4. Points to find There are some specific things that we will find in order to graph a rational function. 1.The y-intercept. Let x = 0 and solve for y. 2.The x-intercepts. Find the zeros of the numerator.

5. Points to find There are some specific things that we will find in order to graph a rational function. 1.The y-intercept. Let x = 0 and solve for y. 2.The x-intercepts. Find the zeros of the numerator. 3.The vertical asymptotes. Find the zeros of the denominator. Check to see if it is a hole.

6. Points to find There are some specific things that we will find in order to graph a rational function. 1.The y-intercept. Let x = 0 and solve for y. 2.The x-intercepts. Find the zeros of the numerator. 3.The vertical asymptotes. Find the zeros of the denominator. Check to see if it is a hole. 4.The horizontal asymptote.

7. Points to find There are some specific things that we will find in order to graph a rational function. 1.The y-intercept. Let x = 0 and solve for y. 2.The x-intercepts. Find the zeros of the numerator. 3.The vertical asymptotes. Find the zeros of the denominator. Check to see if it is a hole. 4.The horizontal asymptote. 5.Do sign analysis.

8. Example 1 Sketch the graph of

9. Example 1 Sketch the graph of When x = 0, y = -3/2.

10. Example 1 Sketch the graph of When x = 0, y = -3/2. No x-intercepts.

11. Example 1 Sketch the graph of When x = 0, y = -3/2. No x-intercepts. The zero of the denominator is 2,so the vertical asymptote is x = 2.

12. Example 1 Sketch the graph of When x = 0, y = -3/2. No x-intercepts. The zero of the denominator is 2,so the vertical asymptote is x = 2. The denominator is a higher power of x than the numerator, so the horizontal asymptote is y = 0.

13. Example 1 Sketch the graph of Do sign analysis by plotting the zeros of the numerator and denominator. ________|_________

14. Example 1 Sketch the graph of (0, -1.5) VA x = 2 HA y = 0 x > 2 + x < 2 --

15. Example 1 Sketch the graph of (0, -1.5) VA x = 2 HA y = 0 x > 2 + x < 2 --

16. Example 2 Sketch the graph of

17. Example 2 Sketch the graph of When x = 0, the denominator is zero, so there is no y-intercept, and x = 0 is a vertical asymptote. When x = ½ the numerator is zero. This is the x-intercept. The horizontal asymptote is y = 2.

18. Example 2 Sketch the graph of (.5, 0) VA x = 0 HA y = 2 ____+____|_____-____|___+____

19. Example 2 Sketch the graph of (.5, 0) VA x = 0 HA y = 2 ____+____|_____-____|___+____

20. Example 3 You try: Sketch the graph of

21. Example 3 You try: Sketch the graph of When x = 0, y = 0. These are both the x and y intercepts. The denominator factors to (x-2)(x+1), so the vertical asymptotes are x = 2 and x = -1. The horizontal asymptote is y = 0. ___-____|__+___|___-____|____+_____ -1 0 2

22. Example 3 You try: Sketch the graph of (0, 0) VA x = 2; x = -1 HA y = 0 __-__|___+___|___-___|___+___ -1 0 2

23. Example 3 You try: Sketch the graph of (0, 0) VA x = 2; x = -1 HA y = 0 __-__|___+___|___-___|___+___ -1 0 2

24. Example 4 You try: Sketch the graph of

25. Example 4 You try: Sketch the graph of When x = 0, y = 3. The zeros of the numerator are + 3. The zeros of the denominator are – 1 and 3, so x = -1 is a vertical asymptote and there is a hole at x = 3. The horizontal asymptote is y = 1. _____|_______|______

26. Example 4 You try: Sketch the graph of (0, 3) (3, 0) (-3, 0) VA x = -1 HA y = 1 Hole x = 3 __+__|__-__|__+__ -3 -1

27. Example 4 You try: Sketch the graph of (0, 3) (-3, 0) VA x = -1 HA y = 1 Hole x = 3 __+__|__-__|__+__ -3 -1

28. Homework Pages 350-351 21-31 odd