1. Check It Out! Example 2 Identify the asymptotes, domain, and range of the function g(x) = – 5. Vertical asymptote: x = 3 Domain: {x|x ≠ 3} Horizontal asymptote: y = –5 Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = 3 and y = –5. 1 x – 3 The value of h is 3. 1 x – (3) g(x) = – 5 h = 3, k = –5. The value of k is –5. Range: {y|y ≠ –5}

2. Some rational functions, including those whose graphs are hyperbolas, have a horizontal asymptote. The existence and location of a horizontal asymptote depends on the degrees of the polynomials that make up the rational function. See what the degree of the numerator is and then what the degree of the denominator is. Larger degree Smaller degree Same degree Smaller degree Larger degree Same degree No HA HA @ y=0 HA @ y= lead coef (improper fraction) (proper fraction) lead coef

4. x+1 The numerator of this type of function is 0 when x = 3 or x = –2. Therefore, the function has x-intercepts at –2 and 3. The denominator of this function is 0 when x = –1. As a result, the graph of the function has a vertical asymptote at the line x = –1.

5. Identify the zeros and vertical asymptotes of f(x) =. TYPE 2: Graphing Rational Functions with Vertical Asymptotes (x 2 + 3x – 4) x + 3 Factor the numerator. (x + 4)(x – 1) x + 3 f(x) = Step 1 Find the zeros and vertical asymptotes. Zeros: X= –4 and 1 The numerator is 0 when x = –4 or x = 1. Vertical asymptote: x = –3 The denominator is 0 when x = –3.

6. Identify the zeros and vertical asymptotes of f(x) =. (x 2 + 3x – 4) x + 3 Step 2 Graph the function. Plot the zeros and draw the asymptote. Then make a table of values to fill in missing points. Vertical asymptote: x = –3 x -11 –8-5-4-3 -215 y -10.5 –7.2 -30 error6 04.5

7. Check It Out! Example 3 Identify the zeros and vertical asymptotes of f(x) =. (x 2 + 7x + 6) x + 3 Factor the numerator. (x + 6)(x + 1) x + 3 f(x) = Step 1 Find the zeros and vertical asymptotes. Zeros: –6 and –1 The numerator is 0 when x = –6 or x = –1. Vertical asymptote: x = –3 The denominator is 0 when x = –3.

8. Check It Out! Example 3 Continued Step 2 Graph the function. Plot the zeros and draw the asymptote. Then make a table of values to fill in missing points. x –7–5–2–1237 y –1.52–404.8610.4 Identify the zeros and vertical asymptotes of f(x) =. (x 2 + 7x + 6) x + 3 Vertical asymptote: x = –3

9. Example 4A: Graphing Rational Functions with Vertical and Horizontal Asymptotes Factor the numerator. Identify the zeros and asymptotes of the function. Then graph. x 2 – 3x – 4 x f(x) = x 2 – 3x – 4 x f(x) = The numerator is 0 when x = 4 or x = –1. The denominator is 0 when x = 0. Degree of p > degree of q. Zeros: 4 and –1 Vertical asymptote: x = 0 Horizontal asymptote: none

10. Example 4A Continued Identify the zeros and asymptotes of the function. Then graph. Vertical asymptote: x = 0

11. Factor the denominator. Identify the zeros and asymptotes of the function. Then graph. x – 2 x2 – 1x2 – 1 f(x) = x – 2 (x – 1)(x + 1) f(x) = The numerator is 0 when x = 2. The denominator is 0 when x = ±1. Degree of p < degree of q. Zero: 2 Vertical asymptote: x = 1, x = –1 Horizontal asymptote: y = 0 Example 4B: Graphing Rational Functions with Vertical and Horizontal Asymptotes

12. Identify the zeros and asymptotes of the function. Then graph. Graph with a graphing calculator or by using a table of values. Example 4B Continued

13. Factor the numerator. Identify the zeros and asymptotes of the function. Then graph. 4x – 12 x – 1x – 1 f(x) = 4(x – 3) x – 1 f(x) = The numerator is 0 when x = 3. The denominator is 0 when x = 1. Zero: 3 Vertical asymptote: x = 1 Horizontal asymptote: y = 4 Example 4C: Graphing Rational Functions with Vertical and Horizontal Asymptotes The horizontal asymptote is y = = = 4. 4 1 leading coefficient of p leading coefficient of q

14. Identify the zeros and asymptotes of the function. Then graph. Graph with a graphing calculator or by using a table of values. Example 4C Continued

15. Check It Out! Example 4a Factor the numerator. Identify the zeros and asymptotes of the function. Then graph. x 2 + 2x – 15 x – 1 f(x) = (x – 3)(x + 5) x – 1 f(x) = The numerator is 0 when x = 3 or x = –5. The denominator is 0 when x = 1. Degree of p > degree of q. Zeros: 3 and –5 Vertical asymptote: x = 1 Horizontal asymptote: none

16. Identify the zeros and asymptotes of the function. Then graph. Check It Out! Example 4a Continued

17. Check It Out! Example 4b Factor the denominator. Identify the zeros and asymptotes of the function. Then graph. x – 2 x 2 + x f(x) = x – 2 x(x + 1) f(x) = The numerator is 0 when x = 2. The denominator is 0 when x = –1 or x = 0. Degree of p < degree of q. Zero: 2 Vertical asymptote: x = –1, x = 0 Horizontal asymptote: y = 0

18. Check It Out! Example 4b Continued Identify the zeros and asymptotes of the function. Then graph.

19. Factor the numerator and the denominator. Identify the zeros and asymptotes of the function. Then graph. 3x 2 + x x2 – 9x2 – 9 f(x) = x(3x – 1) (x – 3) (x + 3) f(x) = The denominator is 0 when x = ±3. Vertical asymptote: x = –3, x = 3 Horizontal asymptote: y = 3 The horizontal asymptote is y = = = 3. 3 1 leading coefficient of p leading coefficient of q Check It Out! Example 4c The numerator is 0 when x = 0 or x = –. 1 3 Zeros: 0 and – 1 3

20. Check It Out! Example 4c Continued Identify the zeros and asymptotes of the function. Then graph.