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Chapter 5

Chapter 5. 5-4 Rational Function. Objectives. Graph rational functions. Transform rational functions by changing parameters. Rational Functions.

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Chapter 5

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  1. Chapter 5 5-4 Rational Function

  2. Objectives Graph rational functions. Transform rational functions by changing parameters.

  3. Rational Functions • rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = 1/x . Its graph is a hyperbola, which has two separate branches. You will learn more about hyperbolas in Chapter 10.

  4. Rational Functions • Like logarithmic and exponential functions, rational functions may have asymptotes. The function f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0

  5. Rational Functions • The rational function f(x) =1/x can be transformed by using methods similar to those used to transform other types of functions.

  6. 1 x + 2 Example 1: Transforming Rational Functions • Using the graph of f(x) = 1/x as a guide, describe the transformation and graph each function. • A. g(x) = Because h = –2, translate f2 units left.

  7. – 3 1 x Example 1 • B. g(x) = Because k = –3, translate f3 units down.

  8. Rational Functions • The values of h and k affect the locations of the asymptotes, the domain, and the range of rational functions whose graphs are hyperbolas

  9. g(x) = – 2 1 1 x – (–3) x + 3 Example 2: Determining Properties of Hyperbolas • Identify the asymptotes, domain, and range of the function g(x) = – 2. Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = –3 and y = –2. Vertical asymptote: x = –3 Domain: {x|x ≠ –3} Horizontal asymptote: y = –2 Range: {y|y ≠ –2}

  10. Rational functions types A discontinuous function is a function whose graph has one or more gaps or breaks. The hyperbola graphed in Example 2 and many other rational functions are discontinuous functions. • A continuous function is a function whose graph has no gaps or breaks. The functions you have studied before this, including linear, quadratic, polynomial, exponential, and logarithmic functions, are continuous functions.

  11. Rational Functions • The graphs of some rational functions are not hyperbolas. Consider the rational function f(x) = (x – 3)(x + 2)/x+1 • and its graph. The numerator of this function is 0 when x = 3 or x = –2. Therefore, the function has x-intercepts at –2 and 3. The denominator of this function is 0 when x = –1. As a result, the graph of the function has a vertical asymptote at the line x = –1.

  12. Vertical Asymptotes

  13. Horizontal Asymptote • Some rational functions, including those whose graphs are hyperbolas, have a horizontal asymptote. The existence and location of a horizontal asymptote depends on the degrees of the polynomials that make up the rational function. • Note that the graph of a rational function can sometimes cross a horizontal asymptote. However, the graph will approach the asymptote when |x| is large.

  14. Horizontal Asymptote

  15. Identify the zerosvertical asymptotes and horizontal asymptote of f(x) = . (x2 + 3x – 4) x+ 3 Example

  16. (x2 + 7x + 6) x+ 3 Example • Identify the zeros,horizontal and vertical asymptotes of f(x) = .

  17. x –2 f(x) = x2–1 GRAPHING • Identify the zeros and asymptotes of the function. Then graph.

  18. HOLES • In some cases, both the numerator and the denominator of a rational function will equal 0 for a particular value of x. As a result, the function will be undefined at this x-value. If this is the case, the graph of the function may have a hole. A hole is an omitted point in a graph.

  19. Example with holes • Identify holes in the graph of f(x)= . Then graph.

  20. Hole at x = 3 solution

  21. Student Guided Practiceand independent practice • Do Problems 11-16 in your book page 345 • Then work on worksheet

  22. Homework • Do even problems from 17-30 in your book page 345

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