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```8.
Classic Cars The data give the estimated value in dollars of a
model of classic car over several years.
15,300
16,100
17,300
18,400
19,600
20,700
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
22,000
a. Find an approximate exponential model for the car’s value
by averaging the successive ratios of the value. Then make
a scatter plot of the data, graph your model with the scatter
plot, and assess its fit to the data.
Let x = 0 represent the year corresponding to the
24
From the point (0, 15.3), a = 15.3.
Estimate b by averaging the value ratios
from year to year:
1.052 + 1.074 + 1.064 + 1.065 + 1.056 + 1.063
6
≈ 1.062
_____
An approximate model is f(x) = 15.3(1.062) .
x
Value (thousand \$)
value \$15,300. Let f(x) be in thousands of dollars.
When modeling a problem in which an exponential
regression equation applies, have students start with
a scatter plot of the data. This will help them
remember that the data points should be increasing
(or decreasing) at a rapid rate, depending on the
original real-world problem.
f(x)
22
20
18
16
x
0
1 2 3 4 5 6 7 8 9 10
The model fits the data very well.
Year
b. In the last year of the data, a car enthusiast spends \$15,100 on a car of the given
model that is in need of some work. The owner then spends \$8300 restoring it.
Use your model to create a table of values with a graphing calculator. How long
does the function model predict the owner should keep the car before it can be
sold for a profit of at least \$5000?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©B.A.E.
Inc./Alamy
The owner spent a total of \$23,400
for the car. To make a profit of at least
\$5000, the selling price must be at
least \$28,400.
The table shows the value exceeding \$28,400 at year 11. Because
the owner bought the car at year 6 of the model, this means that the
model predicts that the owner needs to wait for about 5 years.
Module 14
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788
Lesson 1
1/13/15 12:44 PM
Fitting Exponential Functions to Data
788
9.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
When analyzing a regression equation, ask students
to make sure the equation makes sense with respect
to the original data set. Have them check several data
points in the model to see if the model fits the points
fairly closely. Caution them that the actual values of a
and b in an exponential model of the form f(x) = ab x
may not appear anywhere in the actual data set.
Movies The table shows the average price of a movie ticket in the United States from
2001 to 2010.
Year
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Price (\$)
5.66
5.81
6.03
6.21
6.41
6.55
6.88
7.18
7.50
7.89
a. Make a scatter plot of the data. Then use the first point
and another point on the plot to find an approximate
exponential model for the average ticket price. Then graph
the model with your scatter plot and assess its fit to the data.
Possible answer: Let x represent years after
2001. From the point (0, 5.66), a = 5.66. Choose
the point for 2009, (8, 7.50) and substitute the
coordinates into the model.
f(x) = a ⋅ b x
(_)
An approximate model is f(x) = 5.66(1.036) .
x
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ocean/
Corbis
Overall, the model fits the trend fairly well,
7
6
5
4
middle of the data, and passes below the last
0
x
1 2 3 4 5 6 7 8 9 10 11
Years after 2001
b. Use a graphing calculator to find a regression model for the data, and graph the
model with the scatter plot. How does this model compare to your previous model?
Regression model: f(x) = 5.59(1.037)
x
The regression model is very close to the previous
model. Its graph starts just a bit below the first model’s
graph, and it rises a tiny bit more steeply, but the
models are very close together at the end of the period
of the data.
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Lesson 14.1
f(x)
8
though it overestimates the price a little in the
data point.
Module 14
789
Average price (\$)
9
7.50 = 5.66 ⋅ b 8
1
7.50 _8
=b
5.66
1.036 ≈ b
789
Lesson 1
1/13/15 12:59 PM
What does the regression model predict for the average cost in 2014? How does
this compare with the actual 2014 cost of about \$8.35? A theater owner uses the
model in 2010 to project income for 2014 assuming average sales of 490 tickets
per day at the predicted price. If the actual price is instead \$8.35, did the owner
make a good business decision? Explain.
c.
CONNECT VOCABULARY
For students familiar with the term line of best fit as
used in linear regression models, it is useful to
connect this idea to a notion of “curve of best fit” for
exponential (or quadratic) regression. Modeling these
types of regression models may help them
understand the idea of a “curve of best fit.”
Prediction: f(13) = 5.59(1.037) ≈ \$8.96; this is about \$.616 more per
ticket than the actual. No, for the year, the revenue shortfall would be about
490(365)(0.66) ≈ \$109,000, which is a large amount of money. \$0.61 accounts
for about 7% of the individual ticket price which is a large amount.
13
10. Pharmaceuticals A new medication is being studied to see how quickly it
is metabolized in the body. The table shows how much of an initial dose of
15 milligrams remains in the bloodstream after different intervals of time.
Hours Since
Administration
Amount
Remaining (mg)
0
15
1
14.3
2
13.1
3
12.4
4
11.4
5
10.7
6
10.2
7
9.8
a. Use a graphing calculator to find a regression model. Use the
calculator to graph the model with the scatter plot. How much of the
drug is eliminated each hour?
x
© Houghton Mifflin Harcourt Publishing Company
a. Model: f(x) = 15.0(0.937)
The decay factor is 0.937, so the hourly decay
rate is 1 - 0.937 = 0.063, or 6.3%.
(Answers might vary slightly due to rounding.)
b. The half-life of a drug is how long it takes for half of the drug to be
broken down or eliminated from the bloodstream. Using the Table
function, what is the half-life of the drug to the nearest hour?
b. The half-life is the time to reduce the amount
of medication to 7.5 mg.
The half-life is about 11 hours.
Module 14
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Lesson 1
1/13/15 12:47 PM
Fitting Exponential Functions to Data
790
```
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