21.7: I1.07- Section 5 Part 1

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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: $y=a\cdot{{(x-h)}^{2}}+v$

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: $y=3\cdot{{(x-2)}^{2}}+8$ $y=2.5\cdot{{(x-7)}^{2}}-33.4$ $y=-2\cdot{{(x+3.5)}^{2}}+37$

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
$graph1$	$graph2$	$graph3$	$graph4$
$y=3\cdot{{(x-2)}^{2}}+8$	$y=-1.2\cdot{{(x-2.5)}^{2}}-50$	$y=18\cdot{{(x+3)}^{2}}-158$	$y=3\cdot{{(x+7)}^{2}}-300$

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.

Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution

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