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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



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




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




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




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)




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




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)




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

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