1. Rational Functions (11-2) Objective: Identify excluded values. Identify and use asymptotes to graph rational functions.

2. Rational Functions The function y = 300 / x is an example of a rational function. This function is nonlinear. A rational function can be described by an equation of the form y = p / q, where p and q are polynomials and q ≠ 0. Parent Function: f(x) = 1 / x Type of graph: hyperbola Domain: {x|x ≠ 0} Range: {y|y ≠ 0}

3. Identify Excluded Values Since division by zero is undefined, any value of a variable that results in a denominator of zero in a rational function is excluded from the domain of the function. These are called excluded values for the rational function.

4. Example 1 State the excluded value for each function. a.y = 3 / x  x = 0 b.y = 3 / x+2  x + 2 = 0  x = -2 c.y = 8 / 2x+1  2x + 1 = 0  2x = -1  x = -½

5. Check Your Progress Choose the best answer for the following. A.State the excluded value for the function y = 9.4 / x. A.9.4 B.1 C.0 D.-9.4

6. Check Your Progress Choose the best answer for the following. B.State the excluded value for the function y = 10 / x - 5. A.-5 B.1 C.0 D.5 x – 5 = 0

7. Check Your Progress Choose the best answer for the following. C.State the excluded value for the function y = 5 / 2x + 2. A. 5 / 2 B.-1 C.0 D.-2 2x + 2 = 0 2x = -2

8. Graph Real-Life Rational Functions Depending on the real-world situation, in addition to excluding x-values that make a denominator zero from the domain of a rational function, additional values might have to be excluded from the domain as well.

9. Example 2 If x students will compete in a talent show lasting 100 minutes, the function y = 100 / x represents the number of minutes available for each act. Graph this function. Graph cannot be negative.

10. Check Your Progress Choose the best answer for the following. – Dante and some friends are organizing a lawn service to earn some money for the summer. They have contracted many houses in the neighborhood and are on track to earn $300. The average share of profits y, represented by the function y = 300 / x, decreases as the number of friends x Dante works with. Choose the graph that represents this function. A. B. C. D.

11. Identify and Use Asymptotes In example 2, an excluded value is x = 0. Notice that the graph approaches the vertical line x = 0, but never touches it. The graph also approaches but never touches the horizontal line y = 0. The lines x = 0 and y = 0 are called asymptotes. An asymptote is a line that the graph of a function approaches. A rational function in the form y = a / x – b + c, a ≠ 0, has a vertical asymptote at the x-value that makes the denominator equal zero, x = b. It has a horizontal asymptote at y = c.

12. Identify and Use Asymptotes The domain of y = a / x – b + c is all real numbers except x = b. The range is all real numbers except y = c. Rational functions cannot be traced with a pencil that never leaves the paper, so choose x-values on both sides of the vertical asymptote to graph both portions of the function.

13. Example 3 Identify the asymptotes for each function. Then graph the function. a.y = 3 / x – 4  x = 0  y = -4

14. Example 3 Identify the asymptotes for each function. Then graph the function. b.y = 2 / x + 2  x + 2 = 0  x = -2  y = 0

15. Check Your Progress Choose the best answer for the following. A.Identify the asymptotes of the function y = 6 / x – 2. A.x = 0, y = -2 B.x = -2, y = 2 C.x = -2, y = 0 D.x = 6, y = -2

16. Check Your Progress Choose the best answer for the following. B.Graph the function y = 4 / x – 2. A. B. C. D. x – 2 = 0 x = 2 y = 0