1. Warm-Up 4 minutes Solve each equation. 1) x + 5 = 02) 5x = 03) 5x + 2 = 0 4) x 2 - 5x = 05) x 2 – 5x – 14 = 06) x 3 + 3x 2 – 54x = 0

2. 8.2 Rational Functions and Their Graphs 8.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph

3. Example 1 William begins with 75 milliliters of a 15% acid solution. He adds x milliliters of distilled water to the container holding the acid solution. a) Write a function, C, that represents the acid concentration of the solution in terms of x. 15% of 75 = 11.25 acid solution add x milliliters of distilled water

4. Example 1 William begins with 75 milliliters of a 15% acid solution. He adds x milliliters of distilled water to the container holding the acid solution. b) What is the acid concentration of the solution if 35 milliliters of distilled water is added?

5. Example 2 Find the domain of 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.

6. Vertical Asymptote 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.

7. Example 3 Identify all vertical asymptotes of Factor the denominator. Equations for the vertical asymptotes are x = 2 and x = 1.

8. Homework p.495-496 #5,6,11-15

9. Warm-Up 4 minutes Graph the function. Identify the domain and any vertical asymptotes.

10. 8.2.2 Rational Functions and Their Graphs 8.2.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph

11. 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

12. 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

13. Horizontal Asymptote 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

14. Example 1 Let. Identify all vertical asymptotes and all horizontal asymptotes. 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.

15. Example 2 1 Let. Identify all vertical asymptotes and all horizontal asymptotes. Vertical asymptotes: x = -3 and x = 3 Horizontal asymptotes: leading coefficients numerator and denominator have the same degree y = 2

16. 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).

17. Example 3 Identify all asymptotes and holes in the graph of the rational function. 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

18. Example 4 Use asymptotes to graph the rational function. Write equations for the asymptotes, and graph them as dashed lines. vertical asymptote: x = -4 horizontal asymptote: y = 2 Use a table to help obtain an accurate plot.

19. Homework p.495-496 #17,19,21,25,27,31 USE GRAPH PAPER TO GRAPH THE FUNCTIONS *Only use your calculator to check your answers.