The document describes fitting a quadratic model to a set of (x,y) data points. It shows the process of setting up and solving a system of equations to determine the coefficients a, b, and c in the quadratic function y = ax^2 + bx + c. The system of equations is obtained by setting the quadratic formula equal to the y-value for each data point. The document solves for the coefficients step-by-step to find that the quadratic model for the given data is y = 2x^2 + 2x - 3.

1. SECTION 6-7
The Quadratic Formula
Friday, February 6, 2009

2. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
Friday, February 6, 2009

3. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c

4 = 4a + 2b + c
17 = 9a + 3b + c

Friday, February 6, 2009

4. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c

4 = 4a + 2b + c
17 = 9a + 3b + c

Friday, February 6, 2009

5. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c
 -4 = -4a - 2b - c
4 = 4a + 2b + c
17 = 9a + 3b + c

Friday, February 6, 2009

6. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c
 -4 = -4a - 2b - c
4 = 4a + 2b + c
17 = 9a + 3b + c

Friday, February 6, 2009

7. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c
 -4 = -4a - 2b - c
4 = 4a + 2b + c
13 = 5a + b
17 = 9a + 3b + c

Friday, February 6, 2009

8. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c
4 = 4a + 2b + c
13 = 5a + b
17 = 9a + 3b + c

Friday, February 6, 2009

9. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b
17 = 9a + 3b + c

Friday, February 6, 2009

10. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b
17 = 9a + 3b + c

Friday, February 6, 2009

11. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

Friday, February 6, 2009

12. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b
Friday, February 6, 2009

13. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b
-7 = -3a - b
Friday, February 6, 2009

14. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b
-7 = -3a - b
Friday, February 6, 2009

15. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b
-7 = -3a - b
6 = 2a
Friday, February 6, 2009

16. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b
-7 = -3a - b
6 = 2a
a=3
Friday, February 6, 2009

17. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b
-7 = -3a - b
6 = 2a
a=3
Friday, February 6, 2009

18. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b
-7 = -3a - b 13 = 15 + b
6 = 2a
a=3
Friday, February 6, 2009

19. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b
-7 = -3a - b 13 = 15 + b
6 = 2a b = -2
a=3
Friday, February 6, 2009

20. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b -3 = 3 - 2 + c
-7 = -3a - b 13 = 15 + b
6 = 2a b = -2
a=3
Friday, February 6, 2009

21. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b -3 = 3 - 2 + c
-7 = -3a - b 13 = 15 + b -3 = 1 + c
6 = 2a b = -2
a=3
Friday, February 6, 2009

22. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b -3 = 3 - 2 + c
-7 = -3a - b 13 = 15 + b -3 = 1 + c
6 = 2a b = -2 c = -4
a=3
Friday, February 6, 2009

23. Warm-Up
1. Fit a quadratic model to the following data.
1 2 3 4 5 6
x
-3 4 17 36 61 92
y
−3 = a + b + c 17 = 9a + 3b + c 4 = 4a + 2b + c
 -4 = -4a - 2b - c 3 = -a - b - c
4 = 4a + 2b + c
13 = 5a + b 7 = 3a + b
17 = 9a + 3b + c

13 = 5a + b 13 = 5(3) + b -3 = 3 - 2 + c
-7 = -3a - b 13 = 15 + b -3 = 1 + c y = 3x2 - 2x - 4
6 = 2a b = -2 c = -4
a=3
Friday, February 6, 2009

24. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
Friday, February 6, 2009

25. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0
Friday, February 6, 2009

26. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0
y = 3(0)2 - 2(0) - 4
Friday, February 6, 2009

27. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0
y = 3(0)2 - 2(0) - 4
y=0-0-4
Friday, February 6, 2009

28. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0
y = 3(0)2 - 2(0) - 4
y=0-0-4
y = -4
Friday, February 6, 2009

29. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0
y = 3(0)2 - 2(0) - 4
y=0-0-4
y = -4
y-intercept: (0, -4)
Friday, February 6, 2009

30. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0 x-intercept: When y = 0
y = 3(0)2 - 2(0) - 4
y=0-0-4
y = -4
y-intercept: (0, -4)
Friday, February 6, 2009

31. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0 x-intercept: When y = 0
y = 3(0)2 - 2(0) - 4 0 = 3x2 - 2x - 4
y=0-0-4
y = -4
y-intercept: (0, -4)
Friday, February 6, 2009

32. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0 x-intercept: When y = 0
y = 3(0)2 - 2(0) - 4 0 = 3x2 - 2x - 4
y=0-0-4 4 = 3x2 - 2x
y = -4
y-intercept: (0, -4)
Friday, February 6, 2009

33. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0 x-intercept: When y = 0
y = 3(0)2 - 2(0) - 4 0 = 3x2 - 2x - 4
y=0-0-4 4 = 3x2 - 2x
y = -4 How do we solve for x?
y-intercept: (0, -4)
Friday, February 6, 2009

34. Warm-Up
2. Find the y-intercept of your model. Then find the x-intercept, if you can.
y = 3x2 - 2x - 4
y-intercept: When x = 0 x-intercept: When y = 0
y = 3(0)2 - 2(0) - 4 0 = 3x2 - 2x - 4
y=0-0-4 4 = 3x2 - 2x
y = -4 How do we solve for x?
y-intercept: (0, -4) Complete the square!
Friday, February 6, 2009

35. Quadratic Formula Theorem
Friday, February 6, 2009

36. Quadratic Formula Theorem
If ax2 + bx + c = 0 and a ≠ 0, then
−b ± b 2 − 4ac
x=
2a
Friday, February 6, 2009

37. Example 1
Solve 10x2 - 13x - 3 = 0
Friday, February 6, 2009

38. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac
x=
2a
Friday, February 6, 2009

39. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3)
2
x= =
2(10)
2a
Friday, February 6, 2009

40. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
Friday, February 6, 2009

41. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
13 ± 289
=
20
Friday, February 6, 2009

42. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
13 ±17
13 ± 289
=
=
20
20
Friday, February 6, 2009

43. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
13 ±17
13 ± 289
=
=
20
20
Friday, February 6, 2009

44. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
13 +17
=
20
13 ±17
13 ± 289
=
=
20
20
Friday, February 6, 2009

45. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
13 +17
=
20
13 ±17
13 ± 289
=
=
20 13 −17
20
=
20
Friday, February 6, 2009

46. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
13 +17 30
= =
20 20
13 ±17
13 ± 289
=
=
20 13 −17
20
=
20
Friday, February 6, 2009

47. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
3
13 +17 30
=
= =
2
20 20
13 ±17
13 ± 289
=
=
20 13 −17
20
=
20
Friday, February 6, 2009

48. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
3
13 +17 30
=
= =
2
20 20
13 ±17
13 ± 289
=
=
20 13 −17
20 −4
= =
20 20
Friday, February 6, 2009

49. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
3
13 +17 30
=
= =
2
20 20
13 ±17
13 ± 289
=
=
20 13 −17
20 −4 −1
= = =
20 20 5
Friday, February 6, 2009

50. Example 1
Solve 10x2 - 13x - 3 = 0
2
−b ± b − 4ac 13 ± (−13) − 4(10)(−3) 13 ± 169 +120
2
x= = =
2(10) 20
2a
3
13 +17 30
=
= =
2
20 20
13 ±17
13 ± 289
=
=
20 13 −17
20 −4 −1
= = =
20 20 5
1 3
or
x=− 5 2
Friday, February 6, 2009

51. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
Friday, February 6, 2009

52. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
Friday, February 6, 2009

53. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
Friday, February 6, 2009

54. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−b ± b − 4ac
x=
2a
Friday, February 6, 2009

55. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= =
2(−.005)
2a
Friday, February 6, 2009

56. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
Friday, February 6, 2009

57. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 ± 3.07
=
−.01
Friday, February 6, 2009

58. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 ± 3.07
=
−.01
Friday, February 6, 2009

59. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 + 3.07
=
−2 ± 3.07 −.01
=
−.01
Friday, February 6, 2009

60. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 + 3.07
=
−2 ± 3.07 −.01
=
−2 − 3.07
−.01
=
−.01
Friday, February 6, 2009

61. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 + 3.07
≈ 24.79
=
−2 ± 3.07 −.01
=
−2 − 3.07
−.01
=
−.01
Friday, February 6, 2009

62. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 + 3.07
≈ 24.79
=
−2 ± 3.07 −.01
=
−2 − 3.07
−.01
≈ 375.21
=
−.01
Friday, February 6, 2009

63. Example 2
Pop Fligh’s problem (p. 382 in the book): When is the ball 50 feet high?
50 = -.005x2 + 2x + 3.5
0 = -.005x2 + 2x - 46.5
2
−2 ± 4 − .93
−b ± b − 4ac −2 ± 2 − 4(−.005)(−46.5)
2
x= = =
2(−.005)
2a −.01
−2 + 3.07
≈ 24.79
=
−2 ± 3.07 −.01
=
−2 − 3.07
−.01
≈ 375.21
=
−.01
The ball is about either 24.79 feet or 375.21 feet away.
Friday, February 6, 2009

64. So we need our standard form to use the quadratic
formula. Why is that?
Friday, February 6, 2009

65. Homework
Friday, February 6, 2009

66. Homework
p. 385 #1-26
“It was a high counsel that I once heard given to a young person, ‘Always do
what you are afraid to do.’” - Ralph Waldo Emerson
Friday, February 6, 2009