Curve Fitting with Exponential

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Curve Fitting with Exponential and Logarithmic
Models
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Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Perform a quadratic regression on the
following data
x 1 2 6 11 13
f(x) 3 6 39 120 170
f(x) 0.98x2 0.1x 2.1
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Objectives
Model data by using exponential and logarithmic
functions. Use exponential and logarithmic
models to analyze and predict.
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Vocabulary
exponential regression logarithmic regression
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Analyzing data values can identify a pattern, or
repeated relationship, between two
quantities. Look at this table of values for the
exponential function f(x) 2(3x).
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Notice that the ratio of each y-value and the
previous one is constant. Each value is three
times the one before it, so the ratio of function
values is constant for equally spaced x-values.
This data can be fit by an exponential function
of the form f(x) abx.
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Example 1 Identifying Exponential Data
Determine whether f is an exponential function of
x of the form f(x) abx. If so, find the
constant ratio.
B.
x 1 0 1 2 3
f(x) 2 3 5 8 12
x 1 0 1 2 3
f(x) 16 24 36 54 81
A.
8 12 18 27
1 2 3 4
1 1 1
4 6 9
Ratio
Second differences are constant f is a quadratic
function of x.
This data set is exponential, with a constant
ratio of 1.5.
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Check It Out! Example 1
Determine whether y is an exponential function of
x of the form f(x) abx. If so, find the
constant ratio.
b.
a.
x 1 0 1 2 3
f(x) 2.6 4 6 9 13.5
x 1 0 1 2 3
f(x) 3 2 7 12 17
5 5 5 5
0.66 1 1.5
Ratio
First differences are constant y is a linear
function of x.
This data set is exponential, with a constant
ratio of 1.5.
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In Chapters 2 and 5, you used a graphing
calculator to perform linear progressions and
quadratic regressions to make predictions. You
can also use an exponential model, which is an
exponential function that represents a real data
set.
Once you know that data are exponential, you can
use ExpReg (exponential regression) on your
calculator to find a function that fits. This
method of using data to find an exponential model
is called an exponential regression. The
calculator fits exponential functions to abx, so
translations cannot be modeled.
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Example 2 College Application
Find an exponential model for the data. Use the
model to predict when the tuition at U.T. Austin
will be 6000.
Tuition of the University of Texas Tuition of the University of Texas
Year Tuition
199900 3128
200001 3585
200102 3776
200203 3950
200304 4188
Step 1 Enter data into two lists in a graphing
calculator. Use the exponential regression
feature.
An exponential model is f(x) 3236(1.07t), where
f(x) represents the tuition and t is the number
of years after the 19992000 year.
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Example 2 Continued
Step 2 Graph the data and the function model to
verify that it fits the data.
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Example 2 Continued
Enter 6000 as Y2. Use the intersection feature.
You may need to adjust the dimensions to find the
intersection.
The tuition will be about 6000 when t 9 or
200809.
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Check It Out! Example 2
Use exponential regression to find a function
that models this data. When will the number of
bacteria reach 2000?
Time (min) 0 1 2 3 4 5
Bacteria 200 248 312 390 489 610
Step 1 Enter data into two lists in a graphing
calculator. Use the exponential regression
feature.
An exponential model is f(x) 199(1.25t), where
f(x) represents the tuition and t is the number
of minutes.
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Check It Out! Example 2 Continued
Step 2 Graph the data and the function model to
verify that it fits the data.
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Check It Out! Example 2 Continued
Enter 2000 as Y2. Use the intersection feature.
You may need to adjust the dimensions to find the
intersection.
The bacteria count at 2000 will happen at
approximately 10.3 minutes.
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Many natural phenomena can be modeled by natural
log functions. You can use a logarithmic
regression to find a function
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Example 3 Application
Global Population Growth Global Population Growth
Population (billions) Year
1 1800
2 1927
3 1960
4 1974
5 1987
6 1999
Find a natural log model for the data. According
to the model, when will the global population
exceed 9,000,000,000?
Enter the into the two lists in a graphing
calculator. Then use the logarithmic regression
feature. Press CALC 9LnReg. A logarithmic model
is f(x) 1824 106ln x, where f is the year and
x is the population in billions.
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Example 3 Continued
The calculated value of r2 shows that an equation
fits the data.
Graph the data and function model to verify that
it fits the data.
Use the value feature to find y when x is 9. The
population will exceed 9,000,000,000 in the year
2058.
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Check It Out! Example 4
Use logarithmic regression to find a function
that models this data. When will the speed reach
8.0 m/s?
Time (min) 1 2 3 4 5 6 7
Speed (m/s) 0.5 2.5 3.5 4.3 4.9 5.3 5.6
Enter the into the two lists in a graphing
calculator. Then use the logarithmic regression
feature. Press CALC 9 LnReg. A
logarithmic model is f(x) 0.59 2.64 ln x,
where f is the time and x is the speed.
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Check It Out! Example 4 Continued
The calculated value of r2 shows that an equation
fits the data.
Graph the data and function model to verify that
it fits the data. an equation fits the data.
Use the intersect feature to find y when x is 8.
The time it will reach 8.0 m/s is 16.6 min.
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Lesson Quiz Part I
Determine whether f is an exponential function of
x. If so, find the constant ratio.
x 1 0 1 2
f(x) 10 9 8.1 7.29
1.
yes constant ratio 0.9
x 1 0 1 2 3
f(x) 3 6 12 21 33
2.
no second difference are constant f is
quadratic.
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Lesson Quiz Part II
3. Find an exponential model for the data.
Use the model to estimate when the insurance
value will drop below 2000.
Insurance Value Insurance Value
Year (1990 year 0) Value
0 10,000
2 9,032
5 7,753
9 6,290
11 5,685
f(x) 10,009(0.95)t value will dip below 2000
in year 32 or 2022.