2. Graphing Rational Functions Section 2-6

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Objectives Identify Graph Discontinuities and graph them –Vertical Asymptotes –Horizontal Asymptotes –Slant Asymptotes –Holes Students will be able to determine the Domain and Range of rational functions

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Graph: f(x) = Vertical asymptote: x – 2 = 0 so at x = 2 Dashed line at x = 2 m = 0, n = 1 so m<n HA at y = 0 y-int: (0, -1)

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 f(x) = Vertical Asymptotes: x – 2 = 0 and x + 3 = 0 x = 2, x = -3 M = 0, n = 2 m < n HA at y = 0 y-int: (0, -1) Graph on right

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Factor the numerator and denominator. The factor (x+1) is in both the numerator and denominator, so when cancelled become a “hole” at that point. Since -1 is a common root of both, there is a hole in the graph at -1. Since 2 is a root of the denominator but not the numerator, x = 2 will be a vertical asymptote. Since the polynomials have the same degree, y = 3 will be a horizontal asymptote. Horizontal and Vertical Asymptotes Example: Find all horizontal and vertical asymptotes of f (x) =. y = 3 x = 2 x y

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 A slant asymptote is an asymptote occurs when degree of numerator is exactly 1 more than denominator The slant asymptote is y = 2x – 5. Slant Asymptote Example: Find the slant asymptote for f(x) =. x y x = -3 y = 2x - 5 Divide: Could also use Synthetic Division

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Guidelines for Analyzing Graphs of Rational Functions If f(x) = N(x)/D(x) 1. Simplify f(x) by factoring and reducing 2. When reducing, like factors generate holes 3. Find y-intercept by letting all x’s = 0 4. Solve N(x) = 0 to find x-intercepts 5. Solve D(x) = 0 to find VA 6. Find HA using rules for degree 7. Find SA using synthetic division or long division 8. Plot at least one point on either side of VA 9. Use smooth curves to complete graphs. Do NOT cross asymptotes.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Homework WS 4-6 Quiz next class