1. 3.5 - 1 Rational function A function  of the form where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

2. 3.5 - 2 Rational Function Some examples of rational functions are Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph, that is, a graph that has one or more breaks in it.

3. 3.5 - 3 Domain: (– , 0)  (0,  ) Range: (– , 0)  (0,  ) RECIPROCAL FUNCTION xy – 2– 2– ½– ½ – 1– 1– 1– 1 – ½– ½ – 2– 2 0 undefined ½2 11 2½  decreases on the intervals (– ,0) and (0,  ).

4. 3.5 - 4 Asymptotes Let p(x) and q(x) define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b: 1. If  (x)    as x  a, then the line is a vertical asymptote. 2. If  (x)  b as  x   , then the line y = b is a horizontal asymptote.

5. 3.5 - 5 Determining Asymptotes To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures. 1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.

6. 3.5 - 6 Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (a) If the numerator has lower degree than the denominator, then there is a horizontal asymptote y = 0 (the x- axis).

7. 3.5 - 7 Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (b) If the numerator and denominator have the same degree, and the function is of the form where a n, b n ≠ 0, then the horizontal asymptote has equation

8. 3.5 - 8 Determining Asymptotes 2. Other Asymptotes Determine any other asymptotes. Consider three possibilities: (c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.

9. 3.5 - 9 Steps for Graphing Functions A comprehensive graph of a rational function exhibits these features: 1. all x- and y-intercepts; 2. all asymptotes: vertical, horizontal, and/or oblique; 3. the point at which the graph intersects its nonvertical asymptote (if there is any such point); 4. the behavior of the function on each domain interval determined by the vertical asymptotes and x-intercepts.

10. 3.5 - 10 Graphing a Rational Function Let define a function where p(x) and q(x) are polynomials and the rational expression is written in lowest terms. To sketch its graph, follow these steps. Step 1 Find any vertical asymptotes. Step 2 Find any horizontal or oblique asymptotes. Step 3 Find the y-intercept by evaluating  (0).

11. 3.5 - 11 Graphing a Rational Function Step 4 Find the x-intercepts, if any, by solving  (x) = 0. (These will be the zeros of the numerator, p(x).) Step 5 Determine whether the graph will intersect its nonvertical asymptote y = b or y = mx + b by solving  (x) = b or  (x) = mx + b.

12. 3.5 - 12 Graphing a Rational Function Step 6 Plot selected points, as necessary. Choose an x-value in each domain interval determined by the vertical asymptotes and x-intercepts. Step 7 Complete the sketch.

13. 3.5 - 13 Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE IntervalTest Point Value of  (x) Sign of  (x) Graph Above or Below x-Axis ( – , – 3) – 4– 4 NegativeBelow ( – 3, – 1)– 2– 2 PositiveAbove ( – 1, ½ ) 0NegativeBelow (½,  ) 2PositiveAbove

14. 3.5 - 14 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 1 To find the vertical asymptote(s), solve x 2 + 8x + 16 = 0. Set the denominator equal to 0. Factor. Zero-factor property.

15. 3.5 - 15 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Since the numerator is not 0 when x = – 4, the vertical asymptote has the equation x = – 4. Zero-factor property.

16. 3.5 - 16 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 2 We divide all terms by x 2 to get the equation of the horizontal asymptote. Leading coefficient of numerator Leading coefficient of denominator

17. 3.5 - 17 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 3 The y-intercept is  (0) = – 3/8.

18. 3.5 - 18 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 4 To find the x-intercept(s), if any, we solve  (x) = 0. Set the numerator equal to 0.

19. 3.5 - 19 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 4 Set the numerator equal to 0. Divide by 3. Factor. Zero-factor property The x-intercepts are – 1 and 2.

20. 3.5 - 20 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 5 We set  (x) = 3 and solve to locate the point where the graph intersects the horizontal asymptote. Multiply by x 2 + 8x + 16.

21. 3.5 - 21 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 5 Multiply by x 2 + 8x + 16. Subtract 3x 2. Subtract 24x; add 6. Divide by – 27.

22. 3.5 - 22 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 5 Divide by – 27. The graph intersects its horizontal asymptote at (– 2, 3).

23. 3.5 - 23 Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE Solution Graph Step 6 and 7 Some of the other points that lie on the graph are These are used to complete the graph.

24. 3.5 - 24 Behavior of Graphs of Rational Functions Near Vertical Asymptotes Suppose that  (x) is defined by a rational expression in lowest terms. If n is the largest positive integer such that (x – a) n is a factor of the denominator of  (x), the graph will behave in the manner illustrated.

25. 3.5 - 25 Example 8 GRAPHING A RATIONAL FUNCTION WITH AN OBLIQUE ASYMPTOTE Solution Graph