Asymptotes and Holes Graphing Rational Functions

1 CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 204 Section : Rational Functions Asymptotes and Holes Graphing Rational Functions Asymptotes and Holes Definition of a Rational Function: Definition of a Vertical Asymptote: Definition of a Horizontal Asymptote: SECTION Rational Functions MATH 1330 Precalculus 205 Finding Vertical Asymptotes , Horizontal Asymptotes , and Holes : Example: Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 206 Example: Solution: SECTION Rational Functions MATH 1330 Precalculus 207 Example: Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 208 Example: Solution: SECTION Rational Functions MATH 1330 Precalculus 209 Definition of a Slant Asymptote: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 210 Example: Solution: SECTION Rational Functions MATH 1330 Precalculus 211 Note: For a review of polynomial long division, please refer to Appendix : Dividing Polynomials.

2 Additional Example 1: Solution: The numerator and denominator have no common factors. CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 212 Additional Example 2: SECTION Rational Functions MATH 1330 Precalculus 213 Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 214 Additional Example 3: Solution: SECTION Rational Functions MATH 1330 Precalculus 215 Additional Example 4: Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 216 Additional Example 5: SECTION Rational Functions MATH 1330 Precalculus 217 Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 218 Graphing Rational Functions A Strategy for Graphing Rational Functions : Example: Solution: SECTION Rational Functions MATH 1330 Precalculus 219 CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 220 Additional Example 1: Solution: The numerator and denominator share no common factors.

3 SECTION Rational Functions MATH 1330 Precalculus 221 Additional Example 2: Solution: The numerator and denominator share no common factors. CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 222 SECTION Rational Functions MATH 1330 Precalculus 223 Additional Example 3: Solution: The numerator and denominator share no common factors. CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 224 SECTION Rational Functions MATH 1330 Precalculus 225 Additional Example 4: Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 226 SECTION Rational Functions MATH 1330 Precalculus 227 Additional Example 5: Solution: CHAPTER 2 Polynomial and Rational Functions University of Houston Department of Mathematics 228 Exercise Set : Rational Functions MATH 1330 Precalculus 229 Recall from Section that an even function is symmetric with respect to the y-axis, and an odd function is symmetric with respect to the origin.

4 This can sometimes save time in Graphing Rational Functions . If a function is even or odd, then half of the function can be graphed, and the rest can be graphed using symmetry. Determine if the Functions below are even, odd, or neither. 1. 5()fxx 2. 3()1fxx 3. 24()9fxx 4. 2491()xfxx 5. 21()4xfxx 6. 37()fxx In each of the graphs below, only half of the graph is given. Sketch the remainder of the graph, given that the function is: (a) Even (b) Odd 7. (Notice the Asymptotes at 2x and 0y .) 8. (Notice the Asymptotes at 0x and 0y .) For each of the following graphs: (j) Identify the location of any hole(s) ( removable discontinuities) (k) Identify any x-intercept(s) (l) Identify any y-intercept(s) (m) Identify any vertical asymptote(s) (n) Identify any horizontal asymptote(s) 9. 10. xy xy xy xyExercise Set : Rational Functions University of Houston Department of Mathematics 230 For each of the following Rational Functions : (a) Find the domain of the function (b) Identify the location of any hole(s) ( removable discontinuities) (c) Identify any x-intercept(s) (d) Identify any y-intercept(s) (e) Identify any vertical asymptote(s) (f) Identify any horizontal asymptote(s) (g) Identify any slant asymptote(s) (h) Sketch the graph of the function.

5 Be sure to include all of the above features on your graph. 11. 53)( xxf 12. 74)( xxf 13. xxxf32)( 14. xxxf49)( 15. 36)( xxxf 16. 25)( xxxf 17. 3284)( xxxf 18. 1263)( xxxf 19. )4)(2()3)(2()( xxxxxf 20. )3)(2()6)(3()( xxxxxf 21. 420)(2 xxxxf 22. 5103)(2 xxxxf 23. 324()1xfxx 24. 32()218xfxx 25. )2()2)(53()( xxxxxf 26. )4)(3()75)(4()( xxxxxf 27. 34182)(22 xxxxf 28. 205168)(22 xxxxf 29. 43162xx 30. 32222()4xxxfxx 31. 48)(2 xxf 32. 612)(2 xxxf 33. 1266)(2 xxxxf 34. 152168)(2 xxxxf 35. )2)(4)(1()4)(2)(3()( xxxxxxxf 36. 4595102)(2323 xxxxxxf 37. )3)(1()3)(1)(5()( xxxxxxxf 38. )2)(4()1)(2)(3)(4()( xxxxxxxf 39. 3222918()xxxfxx 40. 423109()xxfxx Exercise Set : Rational Functions MATH 1330 Precalculus 231 Answer the following. 41. In the function 2253323xxfxxx (a) Use the quadratic formula to find the x-intercepts of the function, and then use a calculator to round these answers to the nearest tenth. (b) Use the quadratic formula to find the vertical Asymptotes of the function, and then use a calculator to round these answers to the nearest tenth.

6 42. In the function 2227164xxfxxx (a) Use the quadratic formula to find the x-intercepts of the function, and then use a calculator to round these answers to the nearest tenth. (b) Use the quadratic formula to find the vertical Asymptotes of the function, and then use a calculator to round these answers to the nearest tenth. The graph of a Rational function never intersects a vertical asymptote, but at times the graph intersects a horizontal asymptote. For each function fx below, (a) Find the equation for the horizontal asymptote of the function. (b) Find the x-value where fx intersects the horizontal asymptote. (c) Find the point of intersection of fx and the horizontal asymptote. 43. 22233xxfxxx 44. 2242()7xxfxxx 45. 2223()261xxfxxx 46. 223513xxfxxx 47. 224129()7xxfxxx 48. 2251()5103xxfxxx Answer the following. 49. The function 1266)(2 xxxxf was graphed in Exercise 33. (a) Find the point of intersection of fx and the horizontal asymptote. (b) Sketch the graph of fx as directed in Exercise 33, but also label the intersection of fx and the horizontal asymptote.

7 50. The function 152168)(2 xxxxf was graphed in Exercise 34. (a) Find the point of intersection of fx and the horizontal asymptote. (b) Sketch the graph of fx as directed in Exercise 34, but also label the intersection of fx and the horizontal asymptote.