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Section 5.3 Zeros of the Quadratic 473
Version: Fall 2007
5.3 Exercises
In Exercises 1-8, factor the given qua-
dratic polynomial.
1. x2 + 9x + 14
2. x2 + 6x + 5
3. x2 + 10x + 9
4. x2 + 4x− 21
5. x2 − 4x− 5
6. x2
+ 7x− 8
7. x2 − 7x + 12
8. x2 + 5x− 24
In Exercises 9-16, find the zeros of thegiven quadratic function.
9. f (x) = x2− 2x− 15
10. f (x) = x2 + 4x− 32
11. f (x) = x2 + 10x− 39
12. f (x) = x2 + 4x− 45
13. f (x) = x2− 14x + 40
14. f (x) = x2− 5x− 14
15. f (x) = x2 + 9x− 36
16. f (x) = x2 + 11x− 26
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/1
In Exercises 17-22, perform each of the
following tasks for the quadratic func-tions.
i. Load the function into Y1 of the Y= of your graphing calculator. Adjust thewindow parameters so that the vertexis visible in the viewing window.
ii. Set up a coordinate system on yourhomework paper. Label and scale eachaxis with xmin, xmax, ymin, and ymax.Make a reasonable copy of the imagein the viewing window of your calcu-
lator on this coordinate system andlabel it with its equation.
iii. Use the zero utility on your graph-ing calculator to find the zeros of thefunction. Use these results to plotthe x-intercepts on your coordinatesystem and label them with their co-ordinates.
iv. Use a strictly algebraic technique (nocalculator) to find the zeros of thegiven quadratic function. Show yourwork next to your coordinate system.Be stubborn! Work the problem untilyour algebraic and graphically zerosare a reasonable match.
17. f (x) = x2 + 5x− 14
18. f (x) = x2 + x− 20
19. f (x) = −x2 + 3x + 18
20. f (x) = −x2 + 3x + 40
21. f (x) = x2− 16x− 36
22. f (x) = x2 + 4x− 96
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474 Chapter 5 Quadratic Functions
Version: Fall 2007
In Exercises 23-30, perform each of thefollowing tasks for the given quadraticfunction.
i. Set up a coordinate system on graphpaper. Label and scale each axis. Re-
member to draw all lines with a ruler.
ii. Use the technique of completing thesquare to place the quadratic func-tion in vertex form. Plot the vertexon your coordinate system and labelit with its coordinates. Draw the axisof symmetry on your coordinate sys-tem and label it with its equation.
iii. Use a strictly algebraic technique (nocalculators) to find the x-intercepts
of the graph of the given quadraticfunction. Plot them on your coor-dinate system and label them withtheir coordinates.
iv. Find the y-intercept of the graph of the quadratic function. Plot the y-intercept on your coordinate systemand its mirror image across the axisof symmetry, then label these pointswith their coordinates.
v. Using all the information plotted, drawthe graph of the quadratic functionand label it with the vertex form of its equation. Use interval notation todescribe the domain and range of thequadratic function.
23. f (x) = x2 + 2x− 8
24. f (x) = x2− 6x + 8
25. f (x) = x2 + 4x− 12
26.
f (x) = x
2
+ 8x + 12
27. f (x) = −x2 − 2x + 8
28. f (x) = −x2 − 2x + 24
29. f (x) = −x2 − 8x + 48
30. f (x) = −x2 − 8x + 20
In Exercises 31-38, factor the given qua-dratic polynomial.
31. 42x2 + 5x− 2
32. 3x2 + 7x− 20
33. 5x2 − 19x + 12
34. 54x2 − 3x− 1
35. −4x2 + 9x− 5
36. 3x2 − 5x− 12
37. 2x2 − 3x− 35
38. −6x2 + 25x + 9
In Exercises 39-46, find the zeros of the given quadratic functions.
39. f (x) = 2x2 − 3x− 20
40. f (x) = 2x2 − 7x− 30
41. f (x) = −2x2 + x + 28
42. f (x) = −2x2 + 15x− 22
43. f (x) = 3x2 − 20x + 12
44. f (x) = 4x2 + 11x− 20
45. f (x) = −4x2 + 4x + 15
46. f (x) = −6x2 − x + 12
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Section 5.3 Zeros of the Quadratic 475
Version: Fall 2007
In Exercises 47-52, perform each of thefollowing tasks for the given quadraticfunctions.
i. Load the function into Y1 of the Y= of your graphing calculator. Adjust thewindow parameters so that the vertexis visible in the viewing window.
ii. Set up a coordinate system on yourhomework paper. Label and scale eachaxis with xmin, xmax, ymin, and ymax.Make a reasonable copy of the imagein the viewing window of your calcu-lator on this coordinate system andlabel it with its equation.
iii. Use the zero utility on your graph-
ing calculator to find the zeros of thefunction. Use these results to plotthe x-intercepts on your coordinatesystem and label them with their co-ordinates.
iv. Use a strictly algebraic technique (nocalculator) to find the zeros of thegiven quadratic function. Show yourwork next to your coordinate system.Be stubborn! Work the problem untilyour algebraic and graphically zerosare a reasonable match.
47. f (x) = 2x2 + 3x− 35
48. f (x) = 2x2 − 5x− 42
49. f (x) = −2x2 + 5x + 33
50. f (x) = −2x2 − 5x + 52
51. f (x) = 4x2 − 24x− 13
52. f (x) = 4x2 + 24x− 45
In Exercises 53-60, perform each of thefollowing tasks for the given quadraticfunctions.
i. Set up a coordinate system on graphpaper. Label and scale each axis. Re-
member to draw all lines with a ruler.
ii. Use the technique of completing thesquare to place the quadratic func-tion in vertex form. Plot the vertex
on your coordinate system and labelit with its coordinates. Draw the axisof symmetry on your coordinate sys-tem and label it with its equation.
iii. Use a strictly algebraic method (nocalculators) to find the x-interceptsof the graph of the quadratic func-tion. Plot them on your coordinatesystem and label them with their co-ordinates.
iv. Find the y-intercept of the graph of the quadratic function. Plot the y-
intercept on your coordinate systemand its mirror image across the axisof symmetry, then label these pointswith their coordinates.
v. Using all the information plotted, drawthe graph of the quadratic functionand label it with the vertex form of its equation. Use interval notation todescribe the domain and range of thequadratic function.
53.
f (x) = 2x
2−
8x−
24
54. f (x) = 2x2 − 4x− 6
55. f (x) = −2x2 − 4x + 16
56. f (x) = −2x2 − 16x + 40
57. f (x) = 3x2 + 18x− 48
58. f (x) = 3x2 + 18x− 216
59.
f (x) = 2x
2
+ 10x−
48
60. f (x) = 2x2 − 10x− 100
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476 Chapter 5 Quadratic Functions
Version: Fall 2007
In Exercises 61-66, Use the graph of f (x) = ax2 + bx + c shown to find all so-lutions of the equation f (x) = 0. (Note:Every solution is an integer.)
61.
x
y
5
5
62.
x
y
5
5
63.
x
y
5
5
64.
x
y
5
5
65.
x
y
5
5
66.
x
y
5
5
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Section 5.3 Zeros of the Quadratic 477
Version: Fall 2007
5.3 Answers
1. (x + 2)(x + 7)
3. (x + 9)(x + 1)
5. (x− 5)(x + 1)
7. (x− 4)(x− 3)
9. Zeros: x = −3, x = 5
11. Zeros: x = −13, x = 3
13. Zeros: x = 4, x = 10
15. Zeros: x = −12, x = 3
17.
x
−10
10
y
−30
30f (x)=x2−5x−14
(−
7,0)(−
7,0) (2,0)(2,0)
19.
x−10 10
y
−30
30
f (x)=−x2+3x+18
(−3,0)(−3,0) (6,0)(6,0)
21.
x−10
30
y
−100
100f (x)=x2−16x−36
(−2,0)(−2,0) (18,0)(18,0)
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478 Chapter 5 Quadratic Functions
Version: Fall 2007
23. Domain = (−∞,∞),Range = [−9,∞)
x10
y10 f (x)=(x+1)2−9
x=−
1
(−1,−9)(−1,−9)
(−4,0)(−4,0) (2,0)(2,0)
(0,−8)(0,−8)(−2,−8)(−2,−8)
25. Domain = (−∞,∞),Range = [−16,∞)
x10
y20 f (x)=(x+2)2−16
x=−2
(−2,−16)(−2,−16)
(−6,0)(−6,0) (2,0)(2,0)
(0,−12)(0,−12)(−4,−12)(−4,−12)
27. Domain = (−∞,∞),Range = (−∞, 9]
x10
y20
f (x)=−(x+1)2+9
x=−1
(−1,9)(−1,9)
(−4,0)(−4,0) (2,0)(2,0)
(0,8)(0,8)(−2,8)(−2,8)
29. Domain = (−∞,∞),Range = (−∞, 64]
x20
y100
f (x)=−(x+4)2+64
x=−4
(−4,64)(−4,64)
(−12,0)(−12,0) (4,0)(4,0)
(0,48)(0,48)(−8,48)(−8,48)
31. (7x + 2)(6x− 1)
33. (x− 3)(5x− 4)
35. (4x− 5)(−x + 1)
37. (2x + 7)(x
−
5)
39. Zeros: x = −5/2, x = 4
41. Zeros: x = −7/2, x = 4
43. Zeros: x = 2/3, x = 6
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Section 5.3 Zeros of the Quadratic 479
Version: Fall 2007
45. Zeros: x = −3/2, x = 5/2
47.
x−10
10
y
−50
50f (x)=2x2+3x−35
(−5,0)(−5,0) (3.5,0)(3.5,0)
49.
x−10 10
y
−50
50
f (x)=−2x2+5x+33
(−3,0)(−3,0) (5.5,0)(5.5,0)
51.
x−10 10
y
−100
100f (x)=4x2−24x−13
(−0.5,0)(−0.5,0) (6.5,0)(6.5,0)
53. Domain = (−∞,∞),
Range = [−32,∞)
x10
y50
f (x)=2(x−2)2−32
x=2
(2,−
32)(2,−
32)
(−2,0)(−2,0) (6,0)(6,0)
(0,−24)(0,−24) (4,−24)(4,−24)
55. Domain = (−∞,∞),Range = (−∞, 18]
x10
y20
f (x)=−2(x+1)2+18
x=−1
(−1,18)(−1,18)
(−
4,0)(−
4,0) (2,0)(2,0)
(0,16)(0,16)(−2,16)(−2,16)
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480 Chapter 5 Quadratic Functions
Version: Fall 2007
57. Domain = (−∞,∞),Range = [−75,∞)
x20
y100 f (x)=3(x+3)2−75
x=−3
(−3,−75)(−3,−75)
(−8,0)(−8,0) (2,0)(2,0)
(0,−48)(0,−48)(−6,−48)(−6,−48)
59. Domain = (−∞,∞),Range = [−121/2,∞)
x20
y100 f (x)=2(x+5/2)2−121/2
x=−5/2
(−5/2,−121/2)(−5/2,−121/2)
(−8,0)(−8,0) (3,0)(3,0)
(0,−48)(0,−48)(−5,−48)(−5,−48)
61. −2, 3
63. −3, 0
65. −3, 0