October 07, 2014
2.8 Exploring Data: Quadratic Models
Objective: Use scatter plots and a graphing utility to find quadratic models for data.
Real-life data can sometimes be modeled by a
quadratic equation. Scatter plots and graphing
utilities can be used to help determine whether a linear
model or a quadratic model best fits a particular set of
data.
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Math Guy
Ex1: Decide whether each data set would best be modeled by a
linear model or a quadratic model.
a) (1, 3), (2, 5), (4, 6), (6, 8), (8, 9), (10, 10), (12, 13)
a)
b) (2, 1), (4, 2), (6, 4), (8, 7), (9, 10), (11, 15), (13, 20)
b)
Steps:
1. Make a scatter plot of the data
2. Use the regression feature to
find an equation.
*If you can not tell from the plot, find
both linear and quadratic regression.
Then, compare y-values or r (r2) values
to see which one best fits the data.
LinReg
y=ax+b
a=1.76433121
b=-4.929936306
r2 =.9380417375
r=.9685255482
QuadReg
y=ax 2+bx+c
a=.1328002266
b=-.2183677149
c=.7669396585
R2=.996644016
Reminder: The closer the absolute value of the correlation
coefficient is to 1, the better it fits the data.
LinReg
y=ax+b
a=.8059490085
b=2.763456091
r2=.9715783598
r=.9856867453
October 07, 2014
a)
LinReg
y=ax+b
a=.8059490085
b=2.763456091
r2=.9715783598
r=.9856867453
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QuadReg
y=ax 2+bx+c
a=.0022583397
b=.7771343846
c=2.822703922
R2=.9716501833
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Very close!
Try comparing y-values.
Actual
Values
x
LinReg
QuadReg
1
3
QuadReg
3.5694
3.6021
2
5
4.3754
4.386
4
6
5.9873
5.9674
6
8
7.5992
7.5668
8
9
9.211
9.1843
10
10
10.823
10.82
12
13
12.435
12.474
Ex2: The following table gives the mileage y, in miles per gallon, of a certain
car at various speeds x (in miles per hour).
Use a graphing utility to create a scatter plot of the data.
Use the regression feature of a graphing utility to find a quadratic model that
best fits the data.
Use the model to predict the speed that gives the greatest mileage.
QuadReg
y=ax 2+bx+c
a=-.012023976
b=.9480719281
c=12.28541459