Lesson 6.8 : Fitting Exponential Models to Data

New Concept

This lesson introduces regression analysis. You'll learn to build exponential, logarithmic, and logistic models from real-world data to find the curve that provides the best fit, enabling you to make predictions about future events.

What’s next

Let's put this concept into practice. You'll soon tackle interactive examples and challenge problems, building each type of model from a given data set.

Property

Exponential regression is used to model situations in which growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the command “ExpReg” on a graphing utility to fit an exponential function to a set of data points. This returns an equation of the form,

Note that:

• $b$ must be non-negative.
• when $b > 1$, we have an exponential growth model.
• when $0 < b < 1$, we have an exponential decay model.

Property

Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time. We use the command “LnReg” on a graphing utility to fit a logarithmic function to a set of data points. This returns an equation of the form,

Note that:

• all input values, $x$, must be non-negative.
• when $b > 0$, the model is increasing.
• when $b < 0$, the model is decreasing.

Property

Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command “Logistic” on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form

Note that:

• The initial value of the model is $\frac{c}{1+a}$.
• Output values for the model grow closer and closer to $y = c$ as time increases.