June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or increase in a particular base. For example, let's say a country's population doubles annually. This population growth can be represented in the form of an exponential function.

Exponential functions have many real-world use cases. Mathematically speaking, an exponential function is displayed as f(x) = b^x.

Here we discuss the fundamentals of an exponential function along with relevant examples.

What is the equation for an Exponential Function?

The generic 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 fixed, and x varies

As an illustration, if b = 2, we then 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 a situation where b is greater than 0 and not equal to 1, x will be a real number.

How do you plot Exponential Functions?

To chart an exponential function, we must discover the points where the function crosses the axes. This is known as the x and y-intercepts.

Since the exponential function has a constant, we need to set the value for it. Let's take the value of b = 2.

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

In following this approach, we achieve the domain and the range values for the function. After having the rate, we need to draw 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 larger than 1, the graph will have the below properties:

  • The line passes the point (0,1)

  • The domain is all positive real numbers

  • The range is more than 0

  • The graph is a curved line

  • The graph is increasing

  • The graph is level and constant

  • As x nears negative infinity, the graph is asymptomatic concerning the x-axis

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

In cases where the bases are fractions or decimals between 0 and 1, an exponential function presents with the following properties:

  • The graph passes the point (0,1)

  • The range is greater than 0

  • The domain is entirely 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 approaches negative infinity, the line approaches without bound

  • The graph is flat

  • The graph is unending


There are a few basic rules to bear in mind when working with exponential functions.

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

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

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

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

Rule 3: To grow 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, we are able to write it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is consistently equivalent 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 despite whatever the value of x is.


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

Example 1

Let’s examine the example of the growing of bacteria. If we have a culture of bacteria that duplicates every hour, then at the end of the first hour, we will have twice as many bacteria.

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

At the end of hour three, we will have 8x 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 portray exponential decay. If we have a dangerous substance that decays at a rate of half its amount every hour, then at the end of one hour, we will have half as much substance.

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

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

This can be represented using an exponential equation as follows:

f(t) = 1/2^t

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

As demonstrated, both of these examples pursue a comparable pattern, which is the reason they are able to be depicted using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base stays constant. This indicates that any exponential growth or decline where the base varies is not an exponential function.

For instance, in the matter of compound interest, the interest rate remains the same whilst the base changes in regular time periods.


An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we must enter different values for x and then calculate the matching values for y.

Let's review the following example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As you can see, the worth of y grow very quickly as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like this:

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

Example 2

Draw 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 swiftly as x increases. The reason is because 1/2 is less than 1.

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

The above is a decay function. As shown, the graph is a curved line that descends from right to left and gets smoother as it proceeds.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions display unique 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 terms are the powers of an independent variable figure. The general form of an exponential series is:


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