Activity 4.10 Fitting Curves to Data
Overview:
There are two activities associated with fitting curves to data. Activity 4.10a was designed to be
done in-class and Activity 4.10b was designed to be done as an out-of-class project.
Activity 4.10a asks students to identify the type of function that best describes the data given in a
table. It then asks students to compute the logarithms of the y-values of the given data, to plot
the points (x, ln y), and to find the equation that relates ln y and x. It then asks them to derive the
equation that fits the data represented by (x, y) from the equation that represents (x, ln y).
Activity 4.10b, which was designed as an out-of- class project, asks students to create a scatter
plot of a set of data and to find the regression lines for set of points (x, y), (x, ln y) and (ln x, ln y),
noting the correlation coefficient in each case. It then asks students to derive equations for the
points (x, y) by solving the regression equations found by using (x, ln y) and (ln x, ln y) and to
determine which type of function best represents the data: linear, quadratic, exponential,
logarithmic or power.
Estimated Time Required:
Activity 4.10a should take approximately 30 minutes.
Activity 4.10b should be assigned as an out-of-class project. Completion times may vary.
Technology: A calculator that can produce regression equations.
Prerequisite Concepts:


Using logarithmic scales and linear regression to model data
Exponential and power functions
Discussion:
Review the basic shapes of exponential graph. Consider both exponential growth and decay.
Also review the basic shapes of some simple power functions such as y  x 2 and y  x 3 .
Observe that taking the natural log of both sides of y  Aekx results in ln y = ln A + kx. Because
ln A is a constant, ln y is a linear function of x. The values (x, ln y) provide a good test to
determine if an exponential model of the data is reasonable.
Many students may have difficulty with the fact that ln y is a variable, so you may want to
introduce a dummy variable, such as z, to illustruate the linear relationship z = ln A + kx, but to
get y as a function of x, they must solve for y using ln y = ln A = kx.
For the power function, begin with y  Ax p and take the natural log of both sides:
ln y = ln A + p ln x. Note that the variables are y and x and A and p are constants. Because
when a power function is used to begin with, the two variables ln t and ln y are related by a linear
function. Therefore, if the values of ln y and ln t are linear, a power function will model the
original data.
It is important for students to realize that data may be well fit by more than one type of curve.
Point out that fitting curves to data is often used for making predictions. Though an exponential
function and a power function may both reasonably fit a set of data, they have dramatically
different long-term behaviors, which points out the potential pitfalls in using a model to make
predictions too far into the future.
Activity 4.10a Fitting Curves to Data
1.
Make a scatter plot of the data in the table.
2.
What kind of function appears to describe the behavior of the graph?
3.
Can we find an equation for the function that describes the behavior of the graph? Why or
why not?
y
x
y
1
210
2
405
3
833
4
1590
5
3245
6
6510
x
4.
x
ln(y)
Now fill in the table below, using the values from the table above to calculate the
logarithms:
1
2
3
4
5
6
5.
Plot the points (x, ln(y)) in a scatter plot below:
6.
What type of function appears to describe the behavior of the new graph?
7.
Can you find an equation that relates ln(y) to x?
If so, find it.
8.
ln(y)
Now, can you use the result from
part 7 to find an exponential equation
for the data in part 1? If so, find it.
x
x
4.10b Fitting Curves to Data
Please place your answers in the blanks and on the graph provided. Attach your work to the
back.
At the Kennedy Space Center in Florida, ecologists are concerned about the declining population
of the Florida Scrub-Jay. Below you will find the population data for this threatened bird species.
The population sizes after 1991 are projected values. Normally, projected values would not be
used to develop a mathematical model, but for the purposes of this activity, you will need to treat
these values as if they were real. The years given are the number of years since 1980 (i.e., t = 0
represents 1980).
Year (t)
0
5
9
11
12
13
14
15
16
19
Population Size (N)
3697
2512
2176
2100
1922
1857
1860
1689
1603
1127
I. Construct a scatter plot of the original data (t vs. N).
Using the linear regression function on your calculator, find the line of best fit
(N = mt + b) and the correlation coefficient (r-value).
Line of Best Fit:
Correlation Coefficient:
II. Construct a scatter plot of t vs. ln N.
Using the linear regression function on your calculator, find the line of best fit (ln N = mt + b) and
the correlation coefficient (r-value).
Line of Best Fit:
Correlation Coefficient:
III. Construct a scatter plot of ln t vs. ln N.
Using the linear regression function on your calculator, find the line of best fit
(ln N = m(ln t) + b) and the correlation coefficient (r-value).
Line of Best Fit:
Correlation Coefficient:
IV. Order the three scatter plots (t vs. N, t vs. ln N, ln t vs. ln N) from most linear to least linear.
V. Solve the equations of the two best-fit lines from parts II and III for N and describe the type of
function that resulted (i.e., linear, quadratic, exponential, logarithmic, power).
Part II:
Part III:
VI. On the graph provided below, plot the original data. Also graph the linear function found in
Part I and the two other functions found in Part V. Be sure to number your axes.
Which of the three functions fits the original data the best? Why?
VII. Use the same three functions to predict the population in the year 2100.
1.
2.
3.
Determine which function(s) best fit the data on the basis of the reasonableness of your 2100
predictions. Explain your reasoning.