Alg2 Lesson 9 3
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This document discusses key concepts for graphing rational functions: 1) Vertical and horizontal asymptotes are determined by factors in the denominator that do or do not cancel out. 2) Discontinuities occur where the denominator equals zero, indicating holes. 3) The example function has a vertical asymptote at x= -3, a hole at x=-2, and a horizontal asymptote of y=1.
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Alg2 Lesson 9 3
- 1. Vertical Asymptote Horizontal Asymptote x = 4 y = 1 Hole At x = -2
- 2. Graphing rational functions Factor numerator and denominator Discontinuities occur where denominator equals zero Holes are indicated by factors of denominator that cancel out Vertical asymptotes are indicated by factors of denominator that do not cancel
- 3. Horizontal Asymptotes (predict end behavior) – look at highest degree term of numerator and denominator Degree of top > degree of bottom: no HA Degree of top < degree of bottom: HA is y = 0 Degrees equal: HA is y =
- 4. Example 3-1a Undefined for x = –2 and –3 Hole at x = -2 Vertical asymptote is the line x = -3 Horizontal asymptote: the line y = 1 Determine the equations of any asymptotes and the values of x for any holes in the graph of
- 5. Graph VA: x = -3 HA: y = 1 Hole at x = -2 -2 -2 = -4 (-2, -4)
- 6. Graph x -6 -5 -4 -1 0 1 -8/-3 or 8/3 -7/-2 or 7/2 -6/-1 or 6 -3/2 -2/3 -1/4
- 7. Example 3-2a VA: x = –1 No holes HA: y = 1 Graph
- 8. Example 3-2b x f ( x ) – 4 -4/-3 = 4/3 – 3 -3/-2 = 3/2 – 2 -2/-1 = 2 0 0/1 = 0 1 1/2 2 2/3 3 3/4 Graph
- 9. Example 3-3a Undefined for x = 2 Hole at x = 2 2 (2, 4) x -1 0 1 1 2 3 Graph