Download 5-8 Regression and Quadratic Models

5-8
Curve Fitting with Quadratic Models
For a set of ordered pairs with equally spaced xvalues, a quadratic function has constant
nonzero second differences, as shown below.
Holt Algebra 2
5-8
Curve Fitting with Quadratic Models
Example 1B: Identifying Quadratic Data
Determine whether the data set could represent a
quadratic function. Explain.
x
y
3
4
5
6
7
1
3
9
27
81
Equally spaced x-values
x
y
1st
2nd
Holt Algebra 2
3
4
5
6
7
1
3
9
27
81
2
6
4
18
12
54
36
Find the first and
second differences.
Not a Quadratic
function: second
differences are not
constant for equally
spaced x-values
5-8
Curve Fitting with Quadratic Models
Example 3: Consumer Application
The table shows the cost of circular plastic
wading pools based on the pool’s diameter.
Find a quadratic model for the cost of the pool,
given its diameter. Use the model to estimate
the cost of the pool with a diameter of 8 ft.
Diameter (ft)
Cost
Holt Algebra 2
4
$19.95
5
6
7
$20.25 $25.00 $34.95
5-8
Curve Fitting with Quadratic Models
Example 3 Continued
Step 1 Enter the data
into two lists in a
graphing calculator.
Holt Algebra 2
Step 2 Use the quadratic
regression feature.
5-8
Curve Fitting with Quadratic Models
Example 3 Continued
Step 3 Graph the data
and function model
to verify that the
model fits the data.
Holt Algebra 2
Step 4 Use the table
feature to find the
function value x = 8.
5-8
Curve Fitting with Quadratic Models
Example 3 Continued
A quadratic model is f(x) ≈ 2.4x2 – 21.6x + 67.6,
where x is the diameter in feet and f(x) is the
cost in dollars. For a diameter of 8 ft, the model
estimates a cost of about $49.54.
Holt Algebra 2
5-8
Curve Fitting with Quadratic Models
The table below lists the total estimated numbers of AIDS cases, by year
of diagnosis from 1999 to 2003 in the United States (Source: US Dept. of
Health and Human Services, Centers for Disease Control and Prevention,
HIV/AIDS Surveillance, 2003.)
Use the model to predict the number of AIDS cases in the year 2006.
Year
1999
2000
2001
2002
2003
Holt Algebra 2
AIDS
Cases
41,356
41,267
40,833
41,289
43,171
5-8
Curve Fitting with Quadratic Models
Example 3 Continued
Step 1 Enter the data
into two lists in a
graphing calculator.
Let the year 1998
represent t = 0.
We let 1998 be when t = 0 because it allows
for a more accurate approximation of the
quadratic function. The quadratic
regression’s coefficients of a, b, and c will be
more accurate when there isn’t 2000 units
separating the y-axis and the first data point.
Holt Algebra 2
Step 2 Use the quadratic
regression feature.
5-8
Curve Fitting with Quadratic Models
Example 3 Continued
Step 3 Graph the data
and function model
to verify that the
model fits the data.
Holt Algebra 2
Step 4 Use the table
feature to find the
function value x = 8.
5-8
Curve Fitting with Quadratic Models
Example 3 Continued
A quadratic model is
f(t) ≈ 345.143t2 – 1705.657t + 42903, where x is the
number of years after 1998 and f(t) is the amount of
AIDS cases. For the year 2006, the model estimates
the number of AIDS cases to be 51,347.
Holt Algebra 2
5-8
Curve Fitting with Quadratic Models
HW pg. 377
#12-14,19,24,25,26,36,38,41,50
Holt Algebra 2