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9.7E: Graph Quadratic Functions Using Properties (Exercises)

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Practice Makes Perfect

Exercises 1 - 4: Recognize the Graph of a Quadratic Function

In the following exercises, graph the functions by plotting points.

1. f(x)=x2+3

2. f(x)=x23

3. y=x2+1

4. f(x)=x21

Answer

1.

clipboard_eb78a0f78325e7c8a9cceea709788ca1d.png

3.

clipboard_ef318ed788d73edacb2b69f9a778e9ce9.png

Exercises 5 - 8: Recognize the Graph of a Quadratic Function

For each of the following exercises, determine if the parabola opens up or down.

5. a. f(x)=2x26x7 b. f(x)=6x2+2x+3

6. a. f(x)=4x2+x4 b. f(x)=9x224x16

7. a. f(x)=3x2+5x1 b. f(x)=2x24x+5

8. a. f(x)=x2+3x4 b. f(x)=4x212x9

Answer

5. a. down b. up

7. a. down b. up

Exercises 9 - 12: Find the Axis of Symmetry and Vertex of a Parabola

In the following functions, find

  1. The equation of the axis of symmetry
  2. The vertex of its graph

9. f(x)=x2+8x1

10. f(x)=x2+10x+25

11. f(x)=x2+2x+5

12. f(x)=2x28x3

Answer

9. a. Axis of symmetry: x=4 b. Vertex: (4,17)

11. a. Axis of symmetry: x=1 b. Vertex: (1,2)

Exercises 13 - 24: Find the Intercepts of a Parabola

In the following exercises, find the intercepts of the parabola whose function is given.

13. f(x)=x2+7x+6

14. f(x)=x2+10x11

15. f(x)=x2+8x+12

16. f(x)=x2+5x+6

17. f(x)=x2+8x19

18. f(x)=3x2+x1

19. f(x)=x2+6x+13

20. f(x)=x2+8x+12

21. f(x)=4x220x+25

22. f(x)=x214x49

23. f(x)=x26x9

24. f(x)=4x2+4x+1

Answer

13. y-intercept: (0,6); x-intercept(s): (1,0),(6,0)

15. y-intercept: (0,12); x-intercept(s): (2,0),(6,0)

17. y-intercept: (0,19); x-intercept(s): none

19. y-intercept: (0,13); x-intercept(s): none

21. y-intercept: (0,16); x-intercept(s): (52,0)

23. y-intercept: (0,9); x-intercept(s): (3,0)

Exercises 25 - 42: Graph Quadratic Functions Using Properties

In the following exercises, graph the function by using its properties.

25. f(x)=x2+6x+5

26. f(x)=x2+4x12

27. f(x)=x2+4x+3

28. f(x)=x26x+8

29. f(x)=9x2+12x+4

30. f(x)=x2+8x16

31. f(x)=x2+2x7

32. f(x)=5x2+2

33. f(x)=2x24x+1

34. f(x)=3x26x1

35. f(x)=2x24x+2

36. f(x)=4x26x2

37. f(x)=x24x+2

38. f(x)=x2+6x+8

39. f(x)=5x210x+8

40. f(x)=16x2+24x9

41. f(x)=3x2+18x+20

42. f(x)=2x2+8x10

Answer

25.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).
Figure 9.6.136

27.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).
Figure 9.6.137

29.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.
Figure 9.6.138

31.

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.
Figure 9.6.139

33.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.
Figure 9.6.140

35.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.
Figure 9.6.141

37.

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.
Figure 9.6.142

39.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.
Figure 9.6.143

41.

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.
Figure 9.6.144
Exercises 43 - 48: Solve Maximum and Minimum Applications

In the following exercises, find the maximum or minimum value of each function.

43. f(x)=2x2+x1

44. y=4x2+12x5

45. y=x26x+15

46. y=x2+4x5

47. y=9x2+16

48. y=4x249

Answer

43. The minimum value is 98 when x=14.

45. The maximum value is 6 when x=3.

47. The maximum value is 16 when x=0.

Exercises 49 - 60: Solve Maximum and Minimum Applications

In the following exercises, solve. Round answers to the nearest tenth.

49. An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic function h(t)=16t2+168t+45 find how long it will take the arrow to reach its maximum height, and then find the maximum height.

50. A stone is thrown vertically upward from a platform that is 20 feet height at a rate of 160 ft/sec. Use the quadratic function h(t)=16t2+160t+20 to find how long it will take the stone to reach its maximum height, and then find the maximum height.

51. A ball is thrown vertically upward from the ground with an initial velocity of 109 ft/sec. Use the quadratic function h(t)=16t2+109t+0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

52. A ball is thrown vertically upward from the ground with an initial velocity of 122 ft/sec. Use the quadratic function h(t)=16t2+122t+0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

53. A computer store owner estimates that by charging x dollars each for a certain computer, he can sell 40x computers each week. The quadratic function R(x)=x2+40x is used to find the revenue, R, received when the selling price of a computer is x, Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

54. A retailer who sells backpacks estimates that by selling them for x dollars each, he will be able to sell 100x backpacks a month. The quadratic function R(x)=x2+100x is used to find the R, received when the selling price of a backpack is x. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

55. A retailer who sells fashion boots estimates that by selling them for x dollars each, he will be able to sell 70x boots a week. Use the quadratic function R(x)=x2+70x to find the revenue received when the average selling price of a pair of fashion boots is x. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue per day.

56. A cell phone company estimates that by charging x dollars each for a certain cell phone, they can sell 8x cell phones per day. Use the quadratic function R(x)=x2+8x to find the revenue received per day when the selling price of a cell phone is x. Find the selling price that will give them the maximum revenue per day, and then find the amount of the maximum revenue.

57. A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation A(x)=x(2402x) gives the area of the corral, A, for the length, x, of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral.

58. A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic function A(x)=x(1002x) gives the area, A, of the dog run for the length, x, of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.

59. A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the “walls.” He wants to maximize the area using 80 feet of fencing. The quadratic function A(x)=x(802x) gives the area of the patio, where x is the width of one side. Find the maximum area of the patio.

60. A family of three young children just moved into a house with a yard that is not fenced in. The previous owner gave them 300 feet of fencing to use to enclose part of their backyard. Use the quadratic function A(x)=x(3002x) to determine the maximum area of the fenced in yard.

Answer

49. In 5.3 sec the arrow will reach maximum height of 486 ft.

51. In 3.4 seconds the ball will reach its maximum height of 185.6 feet.

53. 20 computers will give the maximum of $400 in receipts.

55. He will be able to sell 35 pairs of boots at the maximum revenue of $1,225.

57. The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 square feet.

59. The maximum area of the patio is 800 feet.

Exercises 61 - 64: Writing Exercises

61. How do the graphs of the functions f(x)=x2 and f(x)=x21 differ? We graphed them at the start of this section. What is the difference between their graphs? How are their graphs the same?

62. Explain the process of finding the vertex of a parabola.

63. Explain how to find the intercepts of a parabola.

64. How can you use the discriminant when you are graphing a quadratic function?

Answer

1. Answers will vary.

3. Answers will vary.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can recognize the graph of a quadratic equation.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can find the axis of symmetry and vertex of a parabola.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can find the intercepts of a parabola.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can graph quadratic equations in two variables.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve maximum and minimum applications.” “Confidently,” “with some help,” or “No, I don’t get it.”
Figure 9.6.145

b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


This page titled 9.7E: Graph Quadratic Functions Using Properties (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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