1. 2.6 & 2.7 Rational Functions and Their Graphs 2.6 & 2.7 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain and range, write equations for its asymptotes, identify any holes in its graph, and identify the x- and y- intercepts

2. What is a Rational Expression? A rational expression is the quotient of two polynomials. A rational function is a function defined by a rational expression.

3. Simplify

4. Find the Domain Find the domain of To find the domain of a rational function, you 1 st must find the values of x for which the denominator equals 0. x 2 – 9x – 36 = 0 (x – 12)(x + 3) = 0 x = 12 or -3 The domain is all real numbers except 12 and -3.

5. Vertical Asymptote pronounced… “as-im-toht” In a rational function R, if (x – a) is a factor of the denominator but not a factor of the numerator, x = a is vertical asymptote of the graph of R. What is an asymptote? It is a line that a curve approaches but does not reach.

6. To find vertical asymptotes 1.Find the zeros of the denominator 2.Factor numerator 3.Simplify fraction 4.There are vertical asymptotes at any factors that are left in the denominator

7. Identify all vertical asymptotes of Step 1: Factor the denominator. Step 2: Solve the denominator for x. Equations for the vertical asymptotes are x = 2 and x = 1.

8. More Practice Identify the domain and any vertical asymptotes. D: All Real #’s except x=-3,3 VA: at x=-3

9. Look at the table for this function: We can understand why the -3 shows an “error” message. Buy why does the 3 also show an “error” message?

10. That means there is a “Hole” in the graph… That is what happens to the part we “cross off” the fraction. That is where the hole(s) is.

11. Holes in Graphs In a rational function R, if x – b is a factor of the numerator and the denominator, there is a hole in the graph of R when x = b (unless x = b is a vertical asymptote). There is a vertical asymptote at x=-3. And a hole at x=3.

12. Horizontal Asymptote If degree of P < degree of Q, then the horizontal asymptote of R is y = 0. R(x) = is a rational function; P and Q are polynomials P Q So… HA: y=0

13. Horizontal Asymptote R(x) = is a rational function; P and Q are polynomials P Q If degree of P = degree of Q and a and b are the leading coefficients of P and Q, then the horizontal asymptote of R is y =. a b So… HA: y = 1

14. Horizontal Asymptote

15. R(x) = is a rational function; P and Q are polynomials P Q If degree of P > degree of Q, then there is no horizontal asymptote So… HA: D.N.E.

16. Horizontal Asymptotes

17. Slant Asymptote A Slant Asymptote occurs when the degree of the numerator is exactly one degree higher than the degree of the denominator. HA: D.N.E. Therefore: Slant asymptote is Y =

18. Let. Identify the domain and range of the function, all asymptotes and all intercepts. Oh, also are there any holes? Equations for the vertical asymptotes are x = -5 and x = 4. Because the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptotes, but slant asymptote is y = x - 1. D: x ‡ -5, 4 R: ??? ONLY Intercept is ( 0, 0 )

19. 1 Let. Find Domain & Range. Identify all asymptotes, holes and all Intercepts. Vertical asymptotes: x = -3 and x = 3, but NO holes Horizontal asymptotes: leading coefficients numerator and denominator have the same degree y = 2 D: x ‡ 3, -3 R: y ‡ 2 x -intercept: ( ½ √2, 0 ) ( -½ √2, 0 ) Y – intercept: ( 0, 1/9 )

20. Identify all Critical Values in the graph of the rational function, then graph. f(x) = 2x 2 + 2x x 2 – 1 factor: f(x) = 2x(x + 1) (x + 1)(x – 1) hole in the graph: x = –1 vertical asymptote: x = 1 horizontal asymptote: y = 2 D: x ‡ 1, -1 R: y ‡ 2 Intercepts: ( 0, 0 ) ( -1, 0)

21. To graph rational functions 1.Simplify function any restrictions should be listed. 2.Plot y intercept (if any) 3. Plot x intercepts ( zeros of the top) 4.Sketch all asymptotes (dash lines) 5.Plot at least one point between each x intercept and vertical asymptote 6.Use smooth curves to complete graph 7.

22. For, identify all Critical Values, then graph the function. D: Holes: V.A.: H.A.: R: S.A.: X-intercepts: Y-intercepts:

23. For, identify all Critical Values, then graph the function. D: Holes: V.A.: H.A.: R: S.A.: X-intercepts: Y-intercepts:

25. homework p. 152 7-12, 13-18 p. 161 9, 15,23,56,61