I1.07: Section 5 Part 1
Section 5: Using Models.xls to find a quadratic model
If we wish to find a model for the data on the right, this quick scatter plot of the data shows that a straight line will not be sufficient. Such parabolic shapes (similar to the path of a thrown ball) is instead represented mathematically by a quadratic formula, in which the term that contains the input variable x is squared.
There are several ways to write a quadratic formula, all of which can make the same curves. For use in fitting data, the best kind of quadratic formula is y = a (x – h) 2 + v, where the parameters h and v are the x and y coordinates of the vertex of the parabola (that is, its highest or lowest point), and a is a “shape” parameter that determines how sharply (and in which direction) the parabola bends.
The quadratic formula pattern that is most convenient for fitting models: [latex]y=a\cdot{{(x-h)}^{2}}+v[/latex]
a is a “shape” parameter controlling how much (and in which direction) the parabola bends
h is the x coordinate of the vertex (its horizontal distance from the origin)
v is the y coordinate of the vertex (its vertical distance from the origin)
Examples: [latex]y=3\cdot{{(x-2)}^{2}}+8[/latex] [latex]y=2.5\cdot{{(x-7)}^{2}}-33.4[/latex] [latex]y=-2\cdot{{(x+3.5)}^{2}}+37[/latex]
|
| time |
height |
| x |
y |
| 0 |
0.0 |
| 1 |
8.6 |
| 2 |
16.8 |
| 3 |
24.4 |
| 4 |
31.5 |
| 5 |
37.4 |
| 6 |
43.8 |
| 7 |
48.9 |
| 8 |
54.4 |
| 9 |
58.9 |
| 10 |
63.3 |
| 11 |
66.5 |
| 12 |
69.7 |
| 13 |
72.2 |
| 14 |
74.5 |
| 15 |
76.2 |
| 16 |
77.8 |
| 17 |
78.1 |
| 18 |
79.0 |
| 19 |
78.5 |
| 20 |
78.0 |
| 21 |
76.9 |
| 22 |
75.6 |
| 23 |
73.0 |
| 24 |
70.3 |
| 25 |
67.3 |
| 26 |
63.7 |
| 27 |
60.0 |
| 28 |
55.0 |
|
| Examples of graphs of various quadratic formulas |
 |
 |
 |
 |
| [latex]y=3\cdot{{(x-2)}^{2}}+8[/latex] |
[latex]y=-1.2\cdot{{(x-2.5)}^{2}}-50[/latex] |
[latex]y=18\cdot{{(x+3)}^{2}}-158[/latex] |
[latex]y=3\cdot{{(x+7)}^{2}}-300[/latex] |
Example 5: For each of the formulas above, state the location of the vertex of the parabola formed.
Solution: Since the vertex is at (h,v) when a formula is expressed in the form, the coordinates for the vertices are: (2, 8) (2.5, −50) (−3, −158) (−7, −300)
Note that the sign of the x vertex coordinate is the opposite of the sign that the same number has in the formula, since the h value is subtracted when forming the formula.
Quadratic models are somewhat more complicated than linear ones, as is indicated by the fact that a quadratic model has three parameters instead of two. But there is really very little difference in the fitting process from what is done for straight lines: [1] put the data in the appropriate worksheet, [2] spread the C3:E3 formulas down beside the data, [3] make a graph and adjust the vertex (instead of the intercept) and the shape (instead of the slope) until the model and the data match, and [5] write down the formula or use it to predict any values you have been asked for.Licenses & Attributions
CC licensed content, Shared previously
- Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution.
