contoh excel 2
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Exploring the Family of Quadratic Functions
Objective: To examine the vertex and standard forms of the
quadratic function.
Developed as part of a presentation at MCTM 2007 Conference.
Click on the tabs to navigate!
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Vertex Form of Quadratic y = a(x-h)2 + k
a = 1 h = 0 k = 0
x y y = x2
a
-25 625 625 0 51
-24 576 576 0 -50 50
-23 529 529 0 vertex (h, k) axis of symmetry h
-22 484 484 0 h k x y 20
-21 441 441 0 -50 0 -50 625 -20 20
-20 400 400 0 -50 0 k
-19 361 361 0 -50 -625 5000
-18 324 324 0 -50 0 -5000 5000
-17 289 289 0
-16 256 256 0
-15 225 225 0
-14 196 196 0
-13 169 169 0
-12 144 144 0
-11 121 121 0
-10 100 100 0
-9 81 81 0
-8 64 64 0
-7 49 49 0
-6 36 36 0 FALSE FALSE
-5 25 25 0
-4 16 16 0 FALSE y = ax + bx + c
-3 9 9 0
-2 4 4 0 Sinex 2007
-1 1 1 0
y = 0
-800
-600
-400
-200
0
200
400
600
800
-25 -20 -15 -10 -5 0 5 10 15 20 25y
x
show the vertex show the axis of symmetry
See the standard form of the quadratic function
http://academic.pgcc.edu/~ssinexhttp://academic.pgcc.edu/~ssinex -
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0 0 0 0
1 1 1 0
2 4 4 0
3 9 9 04 16 16 0
5 25 25 0
6 36 36 0
7 49 49 0
8 64 64 0
9 81 81 0
10 100 100 0
11 121 121 0
12 144 144 0
13 169 169 0
14 196 196 0
15 225 225 0
16 256 256 0
17 289 289 0
18 324 324 0
19 361 361 0
20 400 400 0
21 441 441 0
22 484 484 0
23 529 529 0
24 576 576 0
25 625 625 0
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Standard Form of Quadratic y = ax2 + bx + c
a = 1 b = 0 c = 0
x y y = x2
-25 625 625 0
-24 576 576 0
-23 529 529 0
-22 484 484 0
-21 441 441 0 roots slope roots
-20 400 400 0 0.00 0 0.2 0.04 x1 = 0.00
-19 361 361 0 0.00 0 0 0
-18 324 324 0 -0.2 0.04 x2 = 0.00
-17 289 289 0
-16 256 256 0
-15 225 225 0 double roots
-14 196 196 0
-13 169 169 0 x = 0
-12 144 144 0 25
-11 121 121 0 slope = 0
-10 100 100 0
-9 81 81 0-8 64 64 0
-7 49 49 0
-6 36 36 0 FALSE y = ax2 + bx + c
-5 25 25 0
-4 16 16 0
-3 9 9 0
-2 4 4 0 Sinex 2007
-1 1 1 0
y = 0
0
100
200
300
400
500
600
700
-25 -20 -15 -10 -5 0 5 10 15 20 25
y
x
See the standard form of the quadratic function
http://academic.pgcc.edu/~ssinexhttp://academic.pgcc.edu/~ssinex -
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0 0 0 0
1 1 1 0
2 4 4 0 Note: You cannot name a variable with the letter "c"!
3 9 9 04 16 16 0
5 25 25 0
6 36 36 0
7 49 49 0
8 64 64 0
9 81 81 0
10 100 100 0
11 121 121 0
12 144 144 0
13 169 169 0
14 196 196 0
15 225 225 0
16 256 256 0
17 289 289 0
18 324 324 0
19 361 361 0
20 400 400 0
21 441 441 0
22 484 484 0
23 529 529 0
24 576 576 0
25 625 625 0
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Looking at Differences y = ax2 + bx + c
First Second a = 1 b = 0 c = 0
x y Dy/Dx Dy/Dx average tim
-25 625 -500 -500 -24.5-24 576 -500 -500 -23.5
-23 529 -500 -500 -22.5
-22 484 -500 -500 -21.5
-21 441 -500 -500 -20.5
-20 400 -500 -500 -19.5
-19 361 -500 -500 -18.5
-18 324 -500 -500 -17.5
-17 289 -500 -500 -16.5-16 256 -500 -500 -15.5
-15 225 -500 -500 -14.5
-14 196 -500 -500 -13.5
-13 169 -500 -500 -12.5
-12 144 -500 -500 -11.5
-11 121 -500 -500 -10.5
-10 100 -500 -500 -9.5
-9 81 -500 -500 -8.5-8 64 -500 -500 -7.5
-7 49 -500 -500 -6.5
-6 36 -500 -500 -5.5
-5 25 -500 -500 FALSE FALSE -4.5
-4 16 -500 -500 -3.5
-3 9 -500 -500 -2.5
-2 4 -500 -500 Sinex 2007 -1.5
-1 1 -500 -500 -0.5
y = x2 + 1E-14x + 6E-13
y = - 500 y = - 500-300
-200
-100
0
100
200
300
-25 -20 -15 -10 -5 0 5 10 15 20 25
y
x
calculate and plot the first difference calculate and plot the second difference
http://academic.pgcc.edu/~ssinexhttp://academic.pgcc.edu/~ssinex -
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0 0 -500 -500 0.5
1 1 -500 -500 1.5
2 4 -500 -500 Note: The first and second differences are plotted against the 2.5
3 9 -500 -500 average time, which is in column L and hidden as white font. 3.54 16 -500 -500 4.5
5 25 -500 -500 5.5
6 36 -500 -500 6.5
7 49 -500 -500 7.5
8 64 -500 -500 8.5
9 81 -500 -500 9.5
10 100 -500 -500 10.5
11 121 -500 -500 11.5
12 144 -500 -500 12.5
13 169 -500 -500 13.5
14 196 -500 -500 14.5
15 225 -500 -500 15.5
16 256 -500 -500 16.5
17 289 -500 -500 17.5
18 324 -500 -500 18.5
19 361 -500 -500 19.5
20 400 -500 -500 20.5
21 441 -500 -500 21.5
22 484 -500 -500 22.5
23 529 -500 -500 23.5
24 576 -500 24.5
25 625
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e
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Baseball Pop-up (travels vertically only) How high will the baseball rise?
Select the astronomical object
initial height = 0 initial velocity = 20 acceleration = 9.81 m/sec2
time, t height, h ho (m) vo (m/sec) a = -g 40 0.00
0.06 1.18
0.12 2.33 object g maximum
0.18 3.44 1 Sun . height
0.24 4.52 1 0 0 0 2 Mercury 3.59 20.39 m
0.3 5.56 1 20.38735 3 Venus 8.87
0.36 6.56 1 0 4 Earth 9.81 velocity
0.42 7.53 1 0 5 - Moon1.62
20.00 m/sec0.48 8.47 6 Mars 3.77 For t=0, initial velocity
0.54 9.37 7 Jupiter 25.95
0.6 10.23 8 Saturn 11.08
0.66 11.06 9 Uranus 10.67
0.72 11.86 10 Neptune . hang time
0.78 12.62 11 Pluto 0.42 4.08 sec
0.84 13.34
0.9 14.03 Can you relate the ho, vo and g to the regression equation? answer
0.96 14.68
1.02 15.30 time step = 0.06 time = 0 0
1.08 15.88 height = 0.00 tracer point
1.14 16.43 Adjust the time step to get the complete
1.2 16.94 time until the baseball hits the ground. ( with constant acceleration due to gravity)
1.26 17.41
1.32 17.85 Sinex 2007
1.38 18.26 g-values: http://aerospaceweb.org/question/astronomy/q0227.shtml
y = -4.905x2 + 20x - 3E-14
0
5
10
15
20
25
0 2 4 6 8
height
time
0
5
10
15
20
25
Path of
Baseball
http://academic.pgcc.edu/~ssinexhttp://academic.pgcc.edu/~ssinex -
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1.44 18.63
1.5 18.96
1.56 19.26
1.62 19.531.68 19.76
1.74 19.95
1.8 20.11
1.86 20.23
1.92 20.32
1.98 20.37
2.04 20.39
2.1 20.37
2.16 20.32
2.22 20.23
2.28 20.10
2.34 19.94
2.4 19.75
2.46 19.52
2.52 19.25
2.58 18.95
2.64 18.61
2.7 18.24
2.76 17.84
2.82 17.39
2.88 16.92
2.94 16.40
3 15.86
3.06 15.27
3.12 14.65
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3.18 14.00
3.24 13.31
3.3 12.58
3.36 11.82
3.42 11.03
3.48 10.20
3.54 9.33
3.6 8.43
3.66 7.49
3.72 6.52
3.78 5.52
3.84 4.47
3.9 3.39
3.96 2.28
4.02 1.13
4.08 -0.05
4.14 -1.27
4.2 -2.52
4.26 -3.81
4.32 -5.14
4.38 -6.504.44 -7.90
4.5 -9.33
4.56 -10.79
4.62 -12.29
4.68 -13.83
4.74 -15.40
4.8 -17.01
4.86 -18.65
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4.92 -20.33
4.98 -22.05
5.04 -23.79
5.1 -25.58
5.16 -27.40
5.22 -29.25
5.28 -31.14
5.34 -33.07
5.4 -35.03
5.46 -37.03
5.52 -39.06
5.58 -41.12
5.64 -43.23
5.7 -45.36
5.76 -47.54
5.82 -49.74
5.88 -51.99
5.94 -54.27
6 -56.58
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velocity20.00 Plot of Velocity against Time
19.41
18.82
18.23
17.65
17.06
16.47
15.88
15.29
14.70
14.11
13.53
12.94
12.35
11.76
11.17 x-intercept = 2.039
10.58 This is where the velocity is equal to zero.
9.99
9.41 This is the same as the first difference on the "differences" tab.
8.82
8.23
7.64
7.05
6.46
y = -9.81x + 20
-50.00
-40.00
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
0 2 4 6 8
velocity
time
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5.87
5.28
4.70
4.11
3.52
2.93
2.34
1.75
1.16
0.58
-0.01
-0.60
-1.19
-1.78
-2.37
-2.96
-3.54
-4.13
-4.72
-5.31
-5.90-6.49
-7.08
-7.66
-8.25
-8.84
-9.43
-10.02
-10.61
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-11.20
-11.78
-12.37
-12.96
-13.55
-14.14
-14.73
-15.32
-15.90
-16.49
-17.08
-17.67
-18.26
-18.85
-19.44
-20.02
-20.61
-21.20
-21.79
-22.38
-22.97-23.56
-24.15
-24.73
-25.32
-25.91
-26.50
-27.09
-27.68
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-28.27
-28.85
-29.44
-30.03
-30.62
-31.21
-31.80
-32.39
-32.97
-33.56
-34.15
-34.74
-35.33
-35.92
-36.51
-37.09
-37.68
-38.27
-38.86