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Properties of Rational Functions and Their Graphs

Properties of Rational Functions and Their Graphs. Rational Function Vertical Asymptotes Horizontal Asymptotes Graphs of Rational Functions. Rational Function.

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Properties of Rational Functions and Their Graphs

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  1. Properties of Rational Functions and Their Graphs • Rational Function • Vertical Asymptotes • Horizontal Asymptotes • Graphs of Rational Functions KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

  2. Rational Function A rational function is a quotient of two polynomials. Therefore a function is a rational function if and only if there is a polynomial and a non-zero polynomial such that The domain of a rational function is the set of all real numbers for which is defined as a real number. Therefore, the domain of consists of all numbers x for which . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  3. Vertical Asymptotes Given the rational function a zero of which is not a zero of represents a vertical asymptote, which is a vertical line , near which the absolute value of the function get larger and larger as x gets closer and closer to . To illustrate, let us look at the plot for the function . Now is a zero of which is not a zero of . Accordingly is a vertical asymptote. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  4. Horizontal Asymptotes • We summarize how to find the horizontal asymptotes of a proper rational function as follows: • The rational function • with has horizontal asymptote • if • if . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  5. Slant Asymptote If is an improper rational function such that degree of degree of then such a rational function does not have a horizontal asymptote. However, since we can write with degree of degree of , then as the absolute value of gets larger gets close to . In case where is linear, it is called a slant or oblique asymptote. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  6. Find the vertical and horizontal asymptotes of the rational function Example 1 • has vertical asymptotes given by the solutions of • Then the vertical asymptotes are: • and • has horizontal asymptote KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  7. Find the vertical and horizontal asymptotes and state if the graph of the rational function intersects the horizontal asymptote or not Example 2 • has vertical asymptotes at • and . • has horizontal asymptote • and since , has solution , the graph of intersects its horizontal asymptote at . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  8. Graphs of Rational Functions Guidelines for graphing rational function The domain of The x and y-intercepts of (if any) The holes, vertical and horizontal asymptotes and point(s) where the graph intersect the horizontal asymptote (if any) Several points on the graph of as needed. Choose an x-value in each domain interval determined by vertical asymptotes and x-intercepts. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  9. Sketch the graph of the rational function Example 3 The domain of consist of all the real numbers except the zeros of denominator. To determine those numbers we need to solve the equation: Therefore the domain of is given by KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  10. Example 3 continue Since is a rational expression, the x-intercepts are the zeros of numerator. Therefore, we need to solve the equation: The only x-intercept is . The other two numbers and are not in the domain of and therefore, do not represent x-intercepts. They–intercept is . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  11. Example 3 continue By factoring the numerator and denominator of For and the graph of has two holes at and . Furthermore, the graph has vertical asymptote and horizontal asymptote . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  12. Example 3 continue Finally to test if the graph crosses the horizontal asymptote, we need to solve the equation Which is a false statement. Therefore, the graph doesn’t cross its horizontal asymptote. We finally obtain some points on the graph. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  13. Example 3 continue Now we can give an accurate sketch of the graph of as: KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

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