June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or rise in a certain base. For example, let us assume a country's population doubles yearly. This population growth can be depicted in the form of an exponential function.

Exponential functions have multiple real-world applications. Mathematically speaking, an exponential function is shown as f(x) = b^x.

In this piece, we will review the basics of an exponential function coupled with appropriate examples.

What is the equation for an Exponential Function?

The general formula for an exponential function is f(x) = b^x, where:

  1. b is the base, and x is the exponent or power.

  2. b is a constant, and x is a variable

As an illustration, if b = 2, then we get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x.

In cases where b is higher than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To graph an exponential function, we must locate the spots where the function crosses the axes. This is known as the x and y-intercepts.

As the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.

To locate the y-coordinates, we need to set the value for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.

According to this approach, we get the range values and the domain for the function. After having the rate, we need to chart them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical properties. When the base of an exponential function is more than 1, the graph would have the below qualities:

  • The line crosses the point (0,1)

  • The domain is all positive real numbers

  • The range is larger than 0

  • The graph is a curved line

  • The graph is on an incline

  • The graph is level and continuous

  • As x approaches negative infinity, the graph is asymptomatic regarding the x-axis

  • As x nears positive infinity, the graph increases without bound.

In instances where the bases are fractions or decimals within 0 and 1, an exponential function displays the following qualities:

  • The graph passes the point (0,1)

  • The range is greater than 0

  • The domain is all real numbers

  • The graph is descending

  • The graph is a curved line

  • As x nears positive infinity, the line in the graph is asymptotic to the x-axis.

  • As x gets closer to negative infinity, the line approaches without bound

  • The graph is smooth

  • The graph is constant


There are several basic rules to bear in mind when engaging with exponential functions.

Rule 1: Multiply exponential functions with an identical base, add the exponents.

For example, if we have to multiply two exponential functions with a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with the same base, subtract the exponents.

For example, if we have to divide two exponential functions with a base of 3, we can compose it as 3^x / 3^y = 3^(x-y).

Rule 3: To increase an exponential function to a power, multiply the exponents.

For example, if we have to raise an exponential function with a base of 4 to the third power, then we can note it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is forever equal to 1.

For example, 1^x = 1 no matter what the rate of x is.

Rule 5: An exponential function with a base of 0 is always equivalent to 0.

For instance, 0^x = 0 regardless of what the value of x is.


Exponential functions are usually utilized to signify exponential growth. As the variable increases, the value of the function increases at a ever-increasing pace.

Example 1

Let’s examine the example of the growth of bacteria. Let us suppose that we have a group of bacteria that multiples by two each hour, then at the close of the first hour, we will have 2 times as many bacteria.

At the end of hour two, we will have 4x as many bacteria (2 x 2).

At the end of hour three, we will have 8 times as many bacteria (2 x 2 x 2).

This rate of growth can be represented using an exponential function as follows:

f(t) = 2^t

where f(t) is the amount of bacteria at time t and t is measured in hours.

Example 2

Similarly, exponential functions can illustrate exponential decay. If we have a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.

At the end of the second hour, we will have a quarter as much substance (1/2 x 1/2).

At the end of the third hour, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as follows:

f(t) = 1/2^t

where f(t) is the amount of substance at time t and t is measured in hours.

As shown, both of these illustrations use a similar pattern, which is why they can be represented using exponential functions.

As a matter of fact, any rate of change can be denoted using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable while the base stays the same. This means that any exponential growth or decline where the base is different is not an exponential function.

For example, in the scenario of compound interest, the interest rate remains the same whereas the base varies in regular intervals of time.


An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we need to plug in different values for x and then measure the equivalent values for y.

Let's look at the following example.

Example 1

Graph the this exponential function formula:

y = 3^x

To start, let's make a table of values.

As shown, the values of y grow very quickly as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:

As seen above, the graph is a curved line that goes up from left to right ,getting steeper as it persists.

Example 2

Plot the following exponential function:

y = 1/2^x

First, let's create a table of values.

As you can see, the values of y decrease very quickly as x surges. This is because 1/2 is less than 1.

Let’s say we were to plot the x-values and y-values on a coordinate plane, it would look like what you see below:

This is a decay function. As you can see, the graph is a curved line that descends from right to left and gets flatter as it goes.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be displayed as f(ax)/dx = ax. All derivatives of exponential functions present special properties by which the derivative of the function is the function itself.

This can be written as following: f'x = a^x = f(x).

Exponential Series

The exponential series is a power series whose terminology are the powers of an independent variable number. The general form of an exponential series is:


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