OBJECTIVES
1. Use a Graphing Utility to Fit an Exponential Function to Data
2. Use a Graphing Utility to Fit a Logarithmic FUnction to Data
3. Use a Graphing Utility to Fit a Logistic Function to Data

Objective 1: Use a Graphing Utility to Fit an Exponential Function to Data

The following data represents the number of bacteria after hours of being infected:

Hours Infected # of Bacteria
0 500
1 597
3 817
5 1153
7 1620
9 2240
15 6160

Using a graphing utility, find the exponential function that best models the data.
1. Input the data into a table in the graphing utility
a. This will plot the points for you
2. Enter y1~a*bx1 into the graphing utility in a separate box
a. read y sub-1 tilde “a” times “b” to the power of “x” sub-1,
3. The graphing will fit an exponential regression
a. It gives the R2 value: a statistical measure of how close the data are to the fitted regression line
i. Between 0 and 1 (0 being no correlation and 1 being a perfect fit)
b. It gives the value for “a” and “b”
c. Use the give “a” and “b” values to write a function that best models the data

Example from the table above:
R2 value: 1
“a” value: 497.928
“b” value: 1.18255

The exponential regression that best fits the data is y=497.928(1.18255)x

Objective 2: Use a Graphing Utility to Fit a Logarithmic Function to Data

The following data represent the population of a small town months after incorporation:

Months After Incorporation Population of the Town
6 3,000
18 4,000
42 5,000
90 6,000
150 7,000

Using a graphing utility, find the logarithmic function that best models the data.
1. Input the data into a table in the graphing utility
a. This will plot the points for you
2. Enter y1~a+b*lnx1 in a separate box
a. read: y sub-1 tilde “a” plus “b” times natural logarithm of “x” sub-1,
3. The graphing will fit a logarithmic regression
a. It gives the R2 value: a statistical measure of how close the data are to the fitted regression line
i. Between 0 and 1 (0 being no correlation and 1 being a perfect fit)
b. It gives the value for “a” and “b”
c. Use the give values to write a function that best models the data

Example from the table above:
R2 value: .983
“a” value: 621.55
“b” value: 1221

The logarithmic regression that best fits the data is y=621.55+1221*ln(x)

Objective 3:Use a Graphing Utility to Fit a Logistic Function to Data

The following data represents the height of a sunflower, in cm, after days of growing.

Day Height (cm)
0 0.00
7 17.93
14 36.36
21 67.76
28 98.10
35 131.00
42 169.50
49 205.50
56 228.30
63 247.10
70 250.50
77 253.80
84 254.50

Using a graphing utility, find the logistic function that best models the data.
1. Input the data into a table in the graphing utility
a. This will plot the points for you
2. Enter y1~a/(1+b*et*x1) in a separate box
a. read: y sub-1 tilde “a” divided by 1 plus “b” times euler’s number to the “t” times “x” sub-1 power
3. The graphing will fit a logistic regression
a. It gives the R2 value: a statistical measure of how close the data are to the fitted regression line
i. Between 0 and 1 (0 being no correlation and 1 being a perfect fit)
b. It gives the value for “a”, “b”, and “t” values
c. Use the give values to write a function that best models the data

Example from the table above:
R2 value: 0.9976
“a” value: 259.963
“b” value: 21.8277
“t” value: -0.0900829
The logistic regression that best fits the data is y=259.963/(1+21.8277e-0.0900829x)