1. Honors Algebra 2 Graphing Rational Functions

2. Quiz 1.State the interval where the function is increasing and decreasing. y=(x+2)2 - 4 Increasing: x greater than -2 Decreasing: x less than -2 Is the function odd, even, or neither 2. y = x2 + 3 3. y = x3 - 1 Even neither

3. Objective - To graph rational functions. The ratio of two polynomial functions is called a rational function. Examples Denominator cannot be zero!

4. A rational function is usually written in the form , where p(x) and q(x) have no common factors.Zeros of a rational function: same zeros as p(x). (a rational expression = 0 where its numerator = 0)

5. What is an asymptote? • An asymptote is a line that a graph approaches as it moves away from the origin. • Asymptotes can be vertical or horizontal. • Vertical asymptotes cannot ever be crossed, but horizontal ones can. • Vertical asymptotes are determined by the zeros of the denominator • Horizontal asymptotes are determined by the degrees of the numerator vs. denominator

6. ASYMPTOTE RULESVertical Asymptotes: located at zeros of q(x).Horizontal Asymptotes: (at most one)a. If degree of the numerator < degree of the denominator: y = 0b. If degree of the numerator = degree of the denominator y = ratio of lead coefficientsc. If degree of the numerator > degree of the denominator no horizontal asymptote**In order to get a good sketch of the graph, you must plot some points between and beyond each x-intercept and vertical asymptote.

7. x = 0 Graph: y = 0

8. x = 0 Graph: y = 0

9. x = 0 Graph: DOMAIN RANGE y = 0 Vertical Asymptotes? Horizontal Asymptotes? X and y intercepts?

10. Graph Vertical asymptote at x = 4 Horizontal asymptote at y = 0 f(x) x graph line x = 4 y -.3 -.3 -.4 -.5 -.7 -1 -2 und. 2 1 .7 .5 -3 -2 -1 0 1 2 3 4 5 6 7 8 x

11. Vertical asymptote at x = 2 and x = -3 Graph Horizontal asymptote at y = 0 f(x) x graph line x = 2 and x = -3 y .3 .4 1 und. -1.5 -1 -1 -1.5 und. 1 .4 .3 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 x

12. Graph Vertical asymptote at x = 2 Horizontal asymptote at y = 0 f(x) x graph line x = 2 y .1 .1 .2 .5 2 und. 2 .5 .2 .1 -3 -2 -1 0 1 2 3 4 5 6 x

13. Identify the holes, VA, x-intercepts, HA and Domain

14. Identify the holes, VA, x-intercepts, HA and Domain

15. Identify the holes, VA, x-intercepts, HA and Domain

16. Analyze the graph of: Vertical asymptote ? Horizontal asymptote ? Domain? Range?

17. Watch Out For Shifts Shift down 1 will change the HA Horizontal Asymptote: Vertical Asymptote: y= -1 x= -3

18. Watch Out For Shifts Shift up 2 will change the HA Horizontal Asymptote: Vertical Asymptote: y= 2 x= 0

19. Check Yourself