Using Linear & Exponential Functions to Model Situations

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Linear and exponential functions can both be used to model data. In this lesson, learn about these two important functions and how and when to use each one.

Using Math to Model the Real World

Mathematical models are used to understand and predict what is going to happen in the future. Scientists take data and find a mathematical function that fits the data as closely as possible, and then they can use that model to predict things they haven't directly observed. For example, meteorologists use mathematical models to predict future weather patterns, and doctors and public health experts use mathematical models to understand and predict the spread of diseases. Two important types of functions that are used to model real-world data are linear functions and exponential functions.

Modeling using Linear Functions

Linear functions are functions which form a straight line when graphed. Many phenomena can be modeled using linear functions. There are two important parts of a linear model, the slope and the y-intercept of the line.

The slope of a line is the rate at which some variable is changing. For a linear model to fit the data, this rate needs to be constant, because a line can only have one slope! The y-intercept is the value of the measured variable at the beginning.

To develop a linear model, use the slope (m) and the y-intercept (b) to write the equation of the line that best fits the data:

y = m x + b

Let's look at an example:

Maddie's car is parked at her school, which is 2.5 miles from her house. If she leaves school and drives in the opposite direction from her house at a rate of 1 mile per minute (this is 60 miles per hour), come up with a model to calculate how far away she is from her house at any time.

In this example, Maddie is driving at a constant rate (1 mile per minute), so a linear model would work well. The time (in minutes) should be plotted on the x-axis and the distance from home should be plotted on the y-axis. The slope of the line will be the rate she is moving, and the y-intercept will be the distance she starts at (2.5 miles from home). This will give you the following linear model for this data:

linear model-1

With this model, Maddie can calculate exactly where she will be at any time after she leaves school. Can you use the model to predict where she will be after 6 minutes?

linear model calculations

Modeling using Exponential Functions

In many cases, real-world situations can be modeled using linear functions, but this is not always true. Sometimes, other functions may fit the data better. Let's look at another example and see how we might model it:

Derrick is studying the growth rate of a certain type of cancer cell. He prepares a flask to grow them in and then carefully places 1,000 cells in the flask. Every 12 hours, he measures the number of cells in the flask. After a few days, he takes all this data and plots on a graph that looks like this:

plotted data

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