1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

2. Chapter 11 Graphing Quadratic Functions, Rational Functions, and Conic Sections

3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 11.3 Graphing Rational Functions by Transformations

4. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Rational functions are quotients of polynomial functions. This means that rational functions can be expresses as where p and q are polynomial functions and q(x) ≠ 0. The domain of a rational function consists of all real numbers except the x-values that make the denominator zero.

5. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the domain of the rational function: Solution The denominator of f(x) = 0 if x = 4. x cannot equal 4. The domain consists of all real numbers except 4.

6. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the domain of the rational function: Solution The denominator of f(x) = 0 if x =  6 or 6. The domain consists of all real numbers except  6 or 6.

7. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall The most basic rational function is the reciprocal function, defined by f(x) = 1/x. The denominator of the reciprocal function is zero when x = 0, so the domain of f is the set of all real numbers except 0.

8. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Locating Vertical Asymptotes Let be a rational function in which p(x) and q(x) have no common factors. If a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.

9. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the vertical asymptotes, if any, of the graph of the rational function. Solution The only zero of the denominator is x = 4. The line x = 4 is the only vertical asymptote of the graph.

10. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the vertical asymptotes, if any, of the graph of the rational function. Solution Factor the denominator. The lines x = 6 and x =  6 are vertical asymptotes. This factor is 0 if x = −6. This factor is 0 if x = 6.

11. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the vertical asymptotes, if any, of the graph of the rational function. Solution Factor the denominator. The graph has no vertical asymptotes. No real numbers make this denominator 0.

12. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

13. Example Find the horizontal asymptotes, if any, of the graph of the rational function. Solution The degree of the numerator, 1, is less than the degree of the denominator, 2. The graph has the x-axis as a horizontal asymptote. The equation of the asymptote is y = 0.

14. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example Find the horizontal asymptotes, if any, of the graph of the rational function. Solution The degree of the numerator, 2, is equal to the degree of the denominator, 2. The leading coefficients of the numerator and denominator, 6 and 1, are used to obtain the equation of the horizontal asymptote. The equation of the asymptote is y = 6/1 or y = 6. continue

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16. Example Use the graph of Solution