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.

graph10

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
graph1 graph2 graph3 graph4
[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.