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# Exponential Regression

Exponential functions are very useful when graphing a wide range of different real-world situations. After all, there are many relationships in our world that are exponential. These include concepts related to finance, population, and many others. That being said, there are different types of exponential functions -- and one type is called exponential regression. But what makes this type of exponential function unique, and what can it teach us about math? When is this function useful? Let's find out:

## Review: Exponential functions

We may recall that the basic equation for an exponential function is ${y}^{x}$. The variable a represents the initial value of the model, and it is always a positive constant. The "outputs" of this function are never equal or less than zero.

A relatively basic exponential graph of ${2}^{x}$ looks like this:

## What is exponential regression?

We use exponential regression to model scenarios in which growth starts slow and then accelerates rapidly without bound. Conversely, exponential regression can also represent scenarios in which decay begins at a rapid pace but slows over time -- becoming closer and closer to zero. The same basic principles of exponential functions apply here.

When expressing exponential regression, we use a slightly different equation: ${y}^{x}$

Note that, unlike our basic exponential function, this function uses an additional variable of b.

With this function, b must be greater than zero and not equal to one. We also know that if b is greater than 1, we are looking at a function of exponential growth. But if b is less than 1, we are looking at a function of exponential decay.

We can also see that by default, this function is well-suited to modeling situations with double the rate of normal exponential growth or decay.

When we use regression, we find an equation of an exponential function that best fits our data. This is useful when we have numerous variables, and we need to describe a relationship between them on a graph. In short, it is a more complex version of an exponential function.

Another important concept here is the "predictive power" of the function. The predictive power is the efficiency with which our model takes errors into account. In other words, it's our "power" to predict results. When we are dealing with exponential regression, we can represent our relative predictive power with R2. This value can vary between 0 and 1. The closer we get to 1, the more accurate our model is.

## Working with exponential regression

Now that we understand how exponential regression works, it's time to start working with these concepts. Consider the following set of data:

• (0,3)
• (1,7)
• (2,10)
• (3,24)
• (4,50)
• (5,95)

Remember, our goal here is to create the equation of the exponential function that best fits our data. Compared to many other operations, this is a bit like "working backward."

Our first step is to enter the x and y-coordinates in our calculator. The easiest way to do this is with a graphing calculator. This allows us to enter both our x and y values, plot our graph, and calculate a function that best fits this dataset very quickly.

When we enter the above values into our graphing calculator, we should be left with this function:

$y=3.0465\left(1.988{x}^{5}\right)$

Now we can plot our graph:

Note that this takes the form of a normal exponential function.

## When is exponential regression a useful concept to know?

Exponential regression is a very useful concept to know in a wide range of situations, as it allows us to model our data effectively. We can also use exponential regression to make accurate predictions about the future. This is where the model's "predictive power" comes into play. Once we have graphed a few of our values and we know that our function is valid, we can use the same formula to simply plug in new values and predict outcomes with high levels of accuracy. In addition, using exponential regression can help us display our data so that other people can easily understand it.

But what are some specific examples of when exponential regression might be useful? One exponential regression model looked at the long-term recovery of patients after they were discharged from the hospital. By using variables that represent the total days of hospitalization for each patient and a prognostic index for long-term recovery, medical professionals were able to accurately predict recovery times for patients. Crucially, they were able to provide patients with this information, letting them know how long they expect to be "out of commission."

In other words, exponential regression can be very useful when we need to relate the concentration of a substance (the response) to elapsed time (the predictor).

## Topics related to the Exponential Regression

Exponential Growth

Exponential Decay

Exponential Functions

## Flashcards covering the Exponential Regression

Algebra 1 Flashcards

College Algebra Flashcards

## Practice tests covering the Exponential Regression

Algebra 1 Diagnostic Tests

College Algebra Diagnostic Tests

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