1. Bellwork 1.Identify any vertical and horizontal asymptotes, or holes in the graphs of the following functions. 2. Write a polynomial function with least degree that has zeros of 2i, -3, and 5.

2. Last Nights Homework 7) A 8) D 9) C 10) E 11) B 12) F 13) a) (-∞,0)U(0,∞)b) VA x = 0, HA y = 0 14) a) (-∞,2)U(2, ∞)b) VA x = 2, HA y = 0 15) a) (-∞,3)U(3,∞)b) VA x = 3, HA y = -1 16) a) (-∞,-1)U(-1,∞) b) VA x = -1, HA y = -5/2 17) a) (-∞,-1)U(-1,1)U(1,∞)b) VA x = 1 x = -1, NO HA 18) a) (-∞,∞)b) NO VA, HA y = 3 19) a) (-∞,-2)U(-2, ∞)b) VA x = -2 c) f(x) -6, -5, -4.5, undefined, -3.5, -3, -2 g(x) -6, -5, -4.5, -4, -3.5, -3, -2 d) f(x) and g(x) are the same except at x = -2. f(x) has a hole at x = -2 40) 2x 3 /(x 2 +x-2) 41) 2x/(x 2 +1)

3. 2.7 Graphs of Rational Functions -How do you sketch graphs of rational functions? How do you decide whether graphs or rational functions have slant asymptotes?

4. Guidelines for Graphing Rational Functions Let f(x) = N(x)/D(x), where N(x) and D(x) are polynomials with no common factors. 1. Find and plot f(0), the y-intercept. 2. Factor the numerator and denominator if necessary. 3. Determine if there are any holes in the graph. 4. Solve N(x) = 0 and plot the x-intercepts. 5. Solve D(x) = 0 to find the vertical asymptotes and sketch 6. Find the horizontal asymptotes and sketch 7. Plot points around x-intercepts and asymptotes. 8. Use smooth lines to draw graph.

5. Example 1: Sketch the graph of the rational functions by hand. 1. g(0) = -3/2(0, -3/2) 2. 3=0 No x-intercepts 3. VA: x-2=0 therefore VA x=2 4. HA: 0<1 therefore HA y = 0 5. (1, -3), (3, 3), (-4, -0.5), (4, 1.5)

6. Example 1: Sketch the graph of the rational functions by hand. 1. f(0)=undefined 2. 2x-1=0(1/2, 0) 3. VA x = 0 4. HA y = 2 5. (-1, 3), (2, 3/2), (-4, -9/4)

7. Example 1: Sketch the graph of the rational functions by hand. 1. f(0)=0 2. x = 0 (0,0) 3. VA x=2, x=-1 4. HA n<m y=0 5. (3, 0,75) (-0.5, 0.2778) (1, -0.5) (-2, -0.5)

8. Example 1: Sketch the graph of the rational functions by hand. 1. f(0) = 9/2(0, 9/2) 2. 0 = 2(x 2 -9) (3,0) (-3,0) 3. VA x=2, x=-2 4. HA y=2 5. (-1, 5.333) (1, 5.33)

9. Slant Asymptotes If the degree of the numerator of a rational function is exactly one more than the degree of its denominator, the graph of the function has a slant (oblique) asymptote. To find the equation of a slant asymptote use long division and graph the resulting quotient without the remainder.

10. Example 2: Graph the rational functions by hand. 1. y-intercept: f(0) = 0(0,0) 2. x-intercept: 0=x 2 -x(0,0) (1,0) 3. VA x=-1 4. HA 2>1NO HA 5. SA (long divide) x-2 6. (-2, -6)

11. Example 2: Graph rational function by hand. 1. f(0)=2 (0, 2) 2. 0=x 2 -x-2 (2,0)(-1,0) 3. VA x=1 4. HA 2>1 NO HA 5. SA y=x

12. Think about it! Write a rational function satisfying the criteria given. 1. Vertical Asymptote: x = -4 Slant Asymptote: y = x – 2 Zero of the function: x = 3 2. Vertical Asymptote: x = 3 Horizontal Asymptote: y = -2 Zero of the function: x = -6

13. Tonight’s Homework Pg 204 #22-28even, #48-54even, 82, 84