June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or rise in a particular base. For instance, let us suppose a country's population doubles every year. This population growth can be represented as an exponential function.

Exponential functions have many real-world applications. In mathematical terms, an exponential function is written as f(x) = b^x.

Here we will learn the fundamentals of an exponential function in conjunction with appropriate examples.

What’s the formula for an Exponential Function?

The common 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

For example, 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 the event where b is larger than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To graph an exponential function, we have to discover the points where the function intersects the axes. This is referred to as the x and y-intercepts.

As the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.

To find the y-coordinates, one must to set the worth for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.

In following this method, we get the domain and the range values for the function. Once we determine the values, 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 characteristics. When the base of an exponential function is larger than 1, the graph will have the below characteristics:

  • The line passes 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 smooth and continuous

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

  • As x advances toward positive infinity, the graph rises without bound.

In events where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following characteristics:

  • The graph intersects the point (0,1)

  • The range is larger than 0

  • The domain is all real numbers

  • The graph is descending

  • The graph is a curved line

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

  • As x advances toward negative infinity, the line approaches without bound

  • The graph is flat

  • The graph is constant


There are some basic rules to remember when dealing with exponential functions.

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

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

Rule 2: To divide exponential functions with an equivalent base, deduct the exponents.

For instance, if we have to divide two exponential functions that posses a base of 3, we can note it as 3^x / 3^y = 3^(x-y).

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

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

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

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

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

For example, 0^x = 0 despite whatever the value of x is.


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

Example 1

Let’s examine the example of the growth of bacteria. Let’s say we have a group of bacteria that doubles hourly, then at the close of hour one, we will have twice as many bacteria.

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

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

This rate of growth can be displayed utilizing an exponential function as follows:

f(t) = 2^t

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

Example 2

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

At the end of hour two, we will have 1/4 as much material (1/2 x 1/2).

After three hours, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be displayed using an exponential equation as below:

f(t) = 1/2^t

where f(t) is the quantity of material at time t and t is assessed in hours.

As shown, both of these samples use a comparable pattern, which is the reason they can be represented using exponential functions.

As a matter of fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable while the base stays fixed. Therefore any exponential growth or decay where the base changes is not an exponential function.

For instance, in the scenario of compound interest, the interest rate remains the same whereas the base changes in regular amounts of time.


An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we need to enter different values for x and then calculate the matching values for y.

Let's review the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As demonstrated, the worth of y grow very rapidly as x rises. Imagine we were to draw this exponential function graph on a coordinate plane, it would look like this:

As seen above, the graph is a curved line that rises from left to right and gets steeper as it persists.

Example 2

Graph the following exponential function:

y = 1/2^x

To begin, let's create a table of values.

As shown, the values of y decrease very swiftly as x surges. This is because 1/2 is less than 1.

If we were to graph the x-values and y-values on a coordinate plane, it would look like the following:

This is a decay function. As shown, the graph is a curved line that gets lower from right to left and gets flatter as it continues.

The Derivative of Exponential Functions

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

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

Exponential Series

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


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