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 specific base. For instance, let us assume a country's population doubles every year. This population growth can be depicted as an exponential function.

Exponential functions have numerous real-world use cases. Expressed mathematically, an exponential function is displayed as f(x) = b^x.

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

What’s 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 a constant, and x varies

For instance, 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 the event where b is greater 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 locate the dots where the function intersects the axes. These are called the x and y-intercepts.

As the exponential function has a constant, it will be necessary 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 technique, we achieve the range values and the domain for the function. After having the worth, we need to graph them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share comparable qualities. When the base of an exponential function is greater than 1, the graph is going to have the following qualities:

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

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

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

  • The graph passes 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 level

  • The graph is constant


There are a few essential rules to remember when dealing with exponential functions.

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

For instance, if we need to multiply two exponential functions that have 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, subtract the exponents.

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

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

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

Rule 4: An exponential function with a base of 1 is always equivalent to 1.

For example, 1^x = 1 no matter what the value 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 commonly utilized to indicate exponential growth. As the variable grows, the value of the function grows at a ever-increasing pace.

Example 1

Let’s observe the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that duplicates hourly, then at the end of the first hour, we will have twice as many bacteria.

At the end of hour two, we will have 4x 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 portrayed an exponential function as follows:

f(t) = 2^t

where f(t) is the number of bacteria at time t and t is measured hourly.

Example 2

Similarly, exponential functions can represent exponential decay. If we have a dangerous material that degenerates at a rate of half its volume every hour, then at the end of hour one, we will have half as much substance.

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

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

This can be represented using an exponential equation as below:

f(t) = 1/2^t

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

As demonstrated, both of these examples follow a similar pattern, which is why they are able to be shown using exponential functions.

As a matter of fact, any rate of change can be indicated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base remains the same. Therefore any exponential growth or decay where the base is different is not an exponential function.

For example, in the scenario of compound interest, the interest rate stays the same whereas the base is static 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 have to enter different values for x and then asses the equivalent values for y.

Let us check out the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As you can see, the values of y grow very fast as x grows. Consider 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 and gets steeper as it persists.

Example 2

Plot the following exponential function:

y = 1/2^x

First, let's draw up a table of values.

As you can see, the values of y decrease very quickly as x rises. 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 is going to look like what you see below:

The above is a decay function. As you can see, the graph is a curved line that decreases from right to left and gets smoother 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 display unique properties where 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 digit. The general form of an exponential series is:


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