June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function measures 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 depicted as an exponential function.

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

In this piece, we will learn the basics of an exponential function along with important 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 varies

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 a situation where b is greater than 0 and does not equal 1, x will be a real number.

How do you graph Exponential Functions?

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

As the exponential function has a constant, we need 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 worth for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

According to this method, we get the range values and the domain for the function. Once we have the worth, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

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

  • The line intersects 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 continuous

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

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

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

  • The graph passes the point (0,1)

  • The range is more than 0

  • The domain is entirely real numbers

  • The graph is decreasing

  • The graph is a curved line

  • As x approaches 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 unending


There are several essential rules to bear in mind when dealing 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 with a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).

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

For instance, 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 instance, if we have to increase 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 equivalent to 1.

For instance, 1^x = 1 regardless of 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 usually leveraged to indicate exponential growth. As the variable increases, the value of the function grows quicker and quicker.

Example 1

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

At the end of the second hour, we will have 4 times 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 displayed an exponential function as follows:

f(t) = 2^t

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

Example 2

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

After two hours, we will have 1/4 as much substance (1/2 x 1/2).

After hour three, 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 demonstrated, both of these samples use a comparable pattern, which is why they can be depicted using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base continues to be the same. Therefore any exponential growth or decline where the base changes is not an exponential function.

For instance, in the scenario of compound interest, the interest rate remains the same while the base is static in normal intervals of time.


An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we must plug in different values for x and then measure the matching values for y.

Let's check out the following example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As demonstrated, the values of y grow very rapidly as x grows. If we were to plot 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 ,getting steeper as it continues.

Example 2

Plot the following exponential function:

y = 1/2^x

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

As you can see, the values of y decrease very swiftly as x surges. The reason 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 is going to look like this:

The above is a decay function. As you can see, the graph is a curved line that gets lower 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 written as f(ax)/dx = ax. All derivatives of exponential functions display unique properties 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 terminology are the powers of an independent variable number. The general form of an exponential series is:


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