Download Graphing and Analyzing Rational Functions: Finding Zeros, Asymptotes, and Intercepts and more Study Guides, Projects, Research Advanced Calculus in PDF only on Docsity!
Section 7.3 Graphing Rational Functions 655
7.3 Exercises
For rational functions Exercises 1 - 20 , follow the Procedure for Graphing Ratio- nal Functions in the narrative, perform- ing each of the following tasks. For rational functions Exercises 1 - 20 , perform each of the following tasks.
i. Set up a coordinate system on graph paper. Label and scale each axis. Re- member to draw all lines with a ruler. ii. Perform each of the nine steps listed in the Procedure for Graphing Ratio- nal Functions in the narrative.
1. f (x) = (x − 3)/(x + 2) 2. f (x) = (x + 2)/(x − 4) 3. f (x) = (5 − x)/(x + 1) 4. f (x) = (x + 2)/(4 − x) 5. f (x) = (2x − 5)/(x + 1) 6. f (x) = (2x + 5/(3 − x) 7. f (x) = (x + 2)/(x^2 − 2 x − 3) 8. f (x) = (x − 3)/(x^2 − 3 x − 4) 9. f (x) = (x + 1)/(x^2 + x − 2) 10. f (x) = (x − 1)/(x^2 − x − 2) 11. f (x) = (x^2 − 2 x)/(x^2 + x − 2) 12. f (x) = (x^2 − 2 x)/(x^2 − 2 x − 8) 13. f (x) = (2x^2 − 2 x − 4)/(x^2 − x − 12) 14. f (x) = (8x − 2 x^2 )/(x^2 − x − 6)
(^1) Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
15. f (x) = (x − 3)/(x^2 − 5 x + 6) 16. f (x) = (2x − 4)/(x^2 − x − 2) 17. f (x) = (2x^2 − x − 6)/(x^2 − 2 x) 18. f (x) = (2x^2 − x − 6)/(x^2 − 2 x) 19. f (x) = (4 + 2x − 2 x^2 )/(x^2 + 4x + 3) 20. f (x) = (3x^2 − 6 x − 9)/(1 − x^2 )
In Exercises 21 - 28 , find the coordinates of the x-intercept(s) of the graph of the given rational function.
21. f (x) =
81 − x^2 x^2 + 10x + 9
22. f (x) =
x − x^2 x^2 + 5x − 6
23. f (x) =
x^2 − x − 12 x^2 + 2x − 3
24. f (x) =
x^2 − 81 x^2 − 4 x − 45
25. f (x) =
6 x − 18 x^2 − 7 x + 12
26. f (x) =
4 x + 36 x^2 + 15x + 54
27. f (x) =
x^2 − 9 x + 14 x^2 − 2 x
28. f (x) =
x^2 − 5 x − 36 x^2 − 9 x + 20
656 Chapter 7 Rational Functions
In Exercises 29 - 36 , find the equations of all vertical asymptotes.
29. f (x) =
x^2 − 7 x x^2 − 2 x
30. f (x) =
x^2 + 4x − 45 3 x + 27
31. f (x) =
x^2 − 6 x + 8 x^2 − 16
32. f (x) =
x^2 − 11 x + 18 2 x − x^2
33. f (x) =
x^2 + x − 12 − 4 x + 12
34. f (x) =
x^2 − 3 x − 54 9 x − x^2
35. f (x) =
16 − x^2 x^2 + 7x + 12
36. f (x) =
x^2 − 11 x + 30 − 8 x + 48
In Exercises 37 - 42 , use a graphing cal- culator to determine the behavior of the given rational function as x approaches both positive and negative infinity by per- forming the following tasks:
i. Load the rational function into the Y= menu of your calculator. ii. Use the TABLE feature of your calcula- tor to determine the value of f (x) for x = 10, 100, 1000, and 10000. Record these results on your homework in ta- ble form. iii. Use the TABLE feature of your calcula- tor to determine the value of f (x) for x = − 10 , − 100 , − 1000 , and − 10000. Record these results on your home- work in table form. iv. Use the results of your tabular explo- ration to determine the equation of
the horizontal asymptote.
37. f (x) = (2x + 3)/(x − 8) 38. f (x) = (4 − 3 x)/(x + 2) 39. f (x) = (4 − x^2 )/(x^2 + 4x + 3) 40. f (x) = (10 − 2 x^2 )/(x^2 − 4) 41. f (x) = (x^2 − 2 x − 3)/(2x^2 − 3 x − 2) 42. f (x) = (2x^2 − 3 x − 5)/(x^2 − x − 6)
In Exercises 43 - 48 , use a purely ana- lytical method to determine the domain of the given rational function. Describe the domain using set-builder notation.
43. f (x) =
x^2 − 5 x − 6 − 9 x − 9
44. f (x) =
x^2 + 4x + 3 x^2 − 5 x − 6
45. f (x) =
x^2 + 5x − 24 x^2 − 3 x
46. f (x) =
x^2 − 3 x − 4 x^2 − 5 x − 6
47. f (x) =
x^2 − 4 x + 3 x − x^2
48. f (x) =
x^2 − 4 x^2 − 9 x + 14
Chapter 7 Rational Functions
Step 9: We can use our graphing calculator to check our graph, as shown in the following sequence of images.
(a) (b) (c)
3. Step 1: The numerator and denominator of
f (x) =
5 − x x + 1
are already factored.
Step 2: Note that x = − 1 makes the denominator zero and is a restriction.
Step 3: Note that the number x = 5 will make the numerator of the rational function f (x) = (5 − x)/(x + 1) equal to zero without making the denominator equal to zero (it is not a restriction). Hence, x = 5 is a zero and (5, 0) will be an x-intercept of the graph of f.
Step 4: Since f is already reduced to lowest terms, we note that the restriction x = − 1 will place a vertical asymptote in the graph of f with equation x = − 1.
Step 5: We will calculate and plot two points, one on each side of the vertical asymp- tote: ( − 2 , − 7) and (0, 5).
Step 6: To find the horizontal asymptote, we use our calculator (see images (a), (b), and (c) below) to determine the end-behavior of f.
(a) (b) (c)
Thus, the line y = − 1 is a horizontal asymptote.
Step 7: Putting all of this information together allows us to draw the following graph.
Section 7.3 Graphing Rational Functions
x
y
x=− 1
y=− 1
(5,0)
Step 8: The restrictions of the reduced form are the same as the restrictions of the original form. Hence, there are no “holes” in the graph.
Step 9: We can use our graphing calculator to check our graph, as shown in the following sequence of images.
(a) (b) (c)
5. Step 1: The numerator and denominator of
f (x) =
2 x − 5 x + 1
are already factored.
Step 2: Note that x = − 1 makes the denominator zero and is a restriction.
Step 3: Note that the number x = 5/ 2 will make the numerator of the rational function f (x) = (2x − 5)/(x + 1) equal to zero without making the denominator equal to zero (it is not a restriction). Hence, x = 5/ 2 is a zero and (5/ 2 , 0) will be an x-intercept of the graph of f.
Step 4: Since f is already reduced to lowest terms, we note that the restriction x = − 1 will place a vertical asymptote in the graph of f with equation x = − 1.
Step 5: We will calculate and plot two points, one on each side of the vertical asymp- tote: ( − 2 , 9) and (0, − 5).
Step 6: To find the horizontal asymptote, we use our calculator (see images (a), (b), and (c) below) to determine the end-behavior of f.
Section 7.3 Graphing Rational Functions
(it is not a restriction). Hence, x = − 2 is a zero and ( − 2 , 0) will be an x-intercept of the graph of f.
Step 4: Since f is already reduced to lowest terms, we note that the restrictions x = − 1 and x = 3 will place vertical asymptotes in the graph of f with equations x = − 1 and x = 3.
Step 5: We will calculate and plot points, one on each side of the vertical asymptote: ( − 1. 5 , 0 .2222), (0, − 0 .6667), and (4, 1 .2). These points are approximations, not exact.
Step 6: To find the horizontal asymptote, we use our calculator (see images (a), (b), and (c) below) to determine the end-behavior of f.
(a) (b) (c)
Thus, the line y = 0 is a horizontal asymptote.
Step 7: Putting all of this information together allows us to draw the following graph.
x
y
(− 2 ,0)
x=− (^1) x=
y=
Step 8: The restrictions of the reduced form are the same as the restrictions of the original form. Hence, there are no “holes” in the graph.
Step 9: We can use our graphing calculator to check our graph, as shown in the following sequence of images. Note that the image in (c) is not very good, but it is enough to convince us that our work above is reasonable.
(a) (b) (c)
Chapter 7 Rational Functions
9. Step 1: Factor numerator and denominator to obtain
f (x) =
x + 1 (x + 2)(x − 1)
Step 2: The restrictions are x = − 2 and x = 1.
Step 3: The number x = − 1 will make the numerator of the rational function f (x) = (x + 1)/((x + 2)(x − 1)) equal to zero without making the denominator equal to zero (it is not a restriction). Hence, x = − 1 is a zero and ( − 1 , 0) will be an x-intercept of the graph of f.
Step 4: Since f is already reduced to lowest terms, we note that the restrictions x = − 2 and x = 1 will place vertical asymptotes in the graph of f with equations x = − 2 and x = 1.
Step 5: We will calculate and plot points, one on each side of the vertical asymptote: ( − 3 , − 0 .5), (0, − 0 .5), and (2, 0 .75). These points are approximations, not exact.
Step 6: To find the horizontal asymptote, we use our calculator (see images (a), (b), and (c) below) to determine the end-behavior of f.
(a) (b) (c)
Thus, a horizontal asymptote is the line y = 0.
Step 7: Putting all of this information together allows us to draw the following graph.
5^ x
y 5
x = − 2 x =
y =0 (( −−^11 ,, 0)0)
Step 8: The restrictions of the reduced form are the same as the restrictions of the original form. Hence, there are no “holes” in the graph.
Chapter 7 Rational Functions
x 10
y 10
x = − 2 x =
y =1 (0(0 ,, 0)0) (2(2 ,, 0)0)
Step 8: The restrictions of the reduced form are the same as the restrictions of the original form. Hence, there are no “holes” in the graph.
Step 9: We can use our graphing calculator to check our graph, as shown in the following sequence of images. Note that the image in (c) is not very good, but it is enough to convince us that our work above is reasonable.
(a) (b) (c)
13. Step 1: Factor.
f (x) =
2 x^2 − 2 x − 4 x^2 − x − 12
2(x^2 − x − 2) x^2 − x − 12
2(x − 2)(x + 1) (x − 4)(x + 3)
Step 2: The restrictions are x = 4 and x = − 3.
Step 3: The numbers x = 2 and x = − 1 will make the numerator of the rational function f equal to zero without making the denominator equal to zero (they are not restrictions). Hence, x = 2 and x = − 1 are zeros and (2, 0) and ( − 1 , 0) will be x- intercepts of the graph of f.
Step 4: Since f is already reduced to lowest terms, we note that the restrictions x = 4 and x = − 3 will place vertical asymptotes in the graph of f with equations x = 4 and x = − 3.
Step 5: We will calculate and plot points, one on each side of the vertical asymptote: ( − 4 , 4 .5), ( − 2 , − 1 .3333), (3, − 1 .3333), and (5, 4 .5). These points are approximations, not exact.
Step 6: To find the horizontal asymptote, we use our calculator (see images (a), (b), and (c) below) to determine the end-behavior of f.
Section 7.3 Graphing Rational Functions
(a) (b) (c)
Thus, the line y = 2 is a horizontal asymptote.
Step 7: Putting all of this information together allows us to draw the following graph.
10^ x
y 10
x = − (^3) x =
y =2 (( −−^11 ,, 0)0)^ (2(2 ,, 0)0)
Step 8: The restrictions of the reduced form are the same as the restrictions of the original form. Hence, there are no “holes” in the graph.
Step 9: We can use our graphing calculator to check our graph, as shown in the following sequence of images.
(a) (b) (c)
15. Step 1: Factor.
f (x) =
x − 3 x^2 − 5 x + 6
x − 3 (x − 3)(x − 2)
Step 2: The restrictions are x = 3 and x = 2.
Step 3: There is no number that will make the numerator zero without making the denominator zero. Hence, the function has no zeros and the graph has no x-intercept.
Step 4: Reduce, obtaining the new function
Section 7.3 Graphing Rational Functions
(a) (b) (c)
17. Step 1: Factor,
f (x) =
2 x^2 − x − 6 x^2 − 2 x
(2x + 3)(x − 2) x(x − 2)
Step 2: The restrictions are x = 0 and x = 2.
Step 3: The number x = − 3 / 2 makes the numerator zero without making the denom- inator zero (it is not a restriction). Hence, x = − 3 / 2 is a zero of f and ( − 3 / 2 , 0) is an x-intercept of the graph of f.
Step 4: Reduce to obtain the new function
g(x) =
2 x + 3 x
Note that x = 0 is still a restriction of the reduced form, so the graph of f must have vertical asymptote with equation x = 0. Note that x = 2 is no longer a restriction of the reduced form. Hence the graph of f will have a “hole” at this restriction. We will deal with this point in a moment.
Step 5: We will calculate and plot points, one on each side of the vertical asymptote: ( − 1 , − 1) and (1, 5).
Step 6: To find the horizontal asymptote, we use our calculator (see images (a), (b), and (c) below) to determine the end-behavior of f.
(a) (b) (c)
Thus, the line y = 2 is a horizontal asymptote.
Step 7: Putting all of this information together allows us to draw the following graph.
Chapter 7 Rational Functions
x 10
y 10
x =
y =
(( −− 33 // 22 ,, 0)0)
(2 , 7 / 2)
Step 8: Recall that we determined that the graph of f will have a “hole” at the restriction x = 2. Use the reduced form g(x) = (2x + 3)/x to determine the y-value of the “hole.”
g(2) =
Hence, there will be a “hole” in the graph of f at (2, 7 /2).
Step 9: We can use our graphing calculator to check our graph, as shown in the following sequence of images.
(a) (b) (c)
19. Step 1: Factor.
f (x) =
4 + 2x − 2 x^2 x^2 + 4x + 3
− 2(x^2 − x − 2) x^2 + 4x + 3
− 2(x − 2)(x + 1) (x + 3)(x + 1)
Step 2: The restrictions are x = − 3 and x = − 1.
Step 3: The number x = 2 makes the numerator zero without making the denominator zero. Hence, x = 2 is a zero of f and (2, 0) is an x-intercept of the graph of f.
Step 4: Reduce, obtaining the new function
g(x) =
− 2(x − 2) x + 3
Note that x = − 3 is still a restriction of the reduced form, so the graph of f must have vertical asymptote with equation x = − 3. Note that x = − 1 is no longer a restriction of the reduced form, so the graph of f will have a “hole” at this restriction. We will deal with the coordinates of this point in a moment.
Chapter 7 Rational Functions
21. x-intercepts occur at values where the numerator of the function is zero, but the denominator is not zero.
81 − x^2 x^2 + 10x + 9
x^2 − 81 x^2 + 10x + 9
(x + 9)(x − 9) (x + 9)(x + 1)
The numerator is zero at x = − 9 and x = 9, and the denominator is zero at x = − 9 and x = − 1. Therefore, the only x-intercept occurs at x = 9.
23. x-intercepts occur at values where the numerator of the function is zero, but the denominator is not zero.
x^2 − x − 12 x^2 + 2x − 3
(x + 3)(x − 4) (x + 3)(x − 1)
The numerator is zero at x = − 3 and x = 4, and the denominator is zero at x = − 3 and x = 1. Therefore, the only x-intercept occurs at x = 4.
25. x-intercepts occur at values where the numerator of the function is zero, but the denominator is not zero.
6 x − 18 x^2 − 7 x + 12
6(x − 3) (x − 3)(x − 4)
The numerator is zero at x = 3, and the denominator is zero at x = 3 and x = 4. Therefore, there are no x-intercepts.
27. x-intercepts occur at values where the numerator of the function is zero, but the denominator is not zero.
x^2 − 9 x + 14 x^2 − 2 x
(x − 2)(x − 7) x(x − 2)
The numerator is zero at x = 2 and x = 7, and the denominator is zero at x = 2 and x = 0. Therefore, the only x-intercept occurs at x = 7.
29. Vertical asymptotes occur where the simplified function is not defined.
x^2 − 7 x x^2 − 2 x
x(x − 7) x(x − 2)
x − 7 x − 2
so the line x = 2 is the only vertical asymptote.
31. Vertical asymptotes occur where the simplified function is not defined.
x^2 − 6 x + 8 x^2 − 16
(x − 4)(x − 2) (x − 4)(x + 4)
x − 2 x + 4
so the line x = − 4 is the only vertical asymptote.
Section 7.3 Graphing Rational Functions
33. Vertical asymptotes occur where the simplified function is not defined.
x^2 + x − 12 − 4 x + 12
(x − 3)(x + 4) − 4(x − 3)
x + 4 4
so there are no vertical asymptotes.
35. Vertical asymptotes occur where the simplified function is not defined.
16 − x^2 x^2 + 7x + 12
x^2 − 16 x^2 + 7x + 12
(x + 4)(x − 4) (x + 4)(x + 3)
x − 4 x + 3
so the line x = − 3 is the only vertical asymptote.
37. Load the equation in (a), then determine the end-behavior in (b) and (c).
(a) (b) (c)
Hence, the equation of the horizontal asymptote is y = 2.
39. Load the equation in (a), then determine the end-behavior in (b) and (c).
(a) (b) (c)
Hence, the equation of the horizontal asymptote is y = − 1.
41. Load the equation in (a), then determine the end-behavior in (b) and (c).
(a) (b) (c)
