Finding an Exponential Regression

1
Finding an Exponential Regression
Use the data in the program file COOL to find an
exponential model.
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Finding an exponential model y
k arx
The data was collected from a hot water bath as
it cooled to an ambient temperature of 27.7
degrees Celcius. One sample was taken every 60
seconds for an hour.
Co
Time sec
3
Finding an exponential model y
k arx
To convert time from seconds to minutes, divide
by 60. Place the cursor on L1. Enter the
formula L1/60 and press ENTER.
To redraw the graph with the new coordinates,
press ZOOMSTAT (Zoom - 9)
4
Finding an exponential model y
k arx
The graph will now be degrees Celcius vs Time in
minutes.
Co
Time min
5
Finding an exponential model y
k arx
Solving Algebraically Step 1 - Find k
The exponential model y k arx will approach
y k as x approaches infinity if 0 lt r lt 1.
Since the ambient room temperature was reported
to be 27.7o C, the water will approach room
temperature and k 27.7o C
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Finding an exponential model y
k arx
Solving Algebraically Step 2 - Find a
If x0, the model y k arx is y k a since
r0 1.
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Finding an exponential model y
k arx
Solving Algebraically Step 3 - Find r
Choose another ordered pair (x, y) and solve for
r.
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Finding an exponential model y
k arx
Solving Algebraically Step 3 - Find r
(cont)
Substituting (50, 43.92) and solving for
r 43.92o 27.7o 60.45o r50
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Finding an exponential model y
k arx
Solving Algebraically Step 4 - Graph
Graph y 27.7 60.45 .9740x
10
Finding an exponential model y
k arx
Using a regression Step 1 - Shift the data to
approach zero
The TI-83 exponential regression is in the form
of y abx which assumes the model will approach
0 as x approaches infinity. Since our data
approaches 27.7, we must shift the data down so
that it will approach 0. After the regression is
complete, the model will be shifted back up the
27.7 degrees.
11
Finding an exponential model y
k arx
Using a regression Step 1 - Shift the data to
approach zero
Using a regression Step 2 - Fit the regression.

12
Finding an exponential model y
k arx
Using a regression Step 2 - Fit the regression.
(cont)
13
Finding an exponential model y
k arx
Using a regression Step 3 - Graph
Using a regression Step 4 - Write a
mathematical model
ToC 27.7oC 54.24oC .9758 Tmin
Using a regression Step 4 - Explain the
mathematical model
The ambient temperature is 27.7o. The initial
temperature is 54.24o greater than the ambient
temperature and retains 97.58 of its temperature
each minute or it decreases its temperature 2.42
each minute.