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 suppose a country's population doubles yearly. This population growth can be depicted as an exponential function.

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

Today we will review the essentials of an exponential function along with appropriate examples.

What is the equation for an Exponential Function?

The common equation 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, 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 graph Exponential Functions?

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

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

To find the y-coordinates, its essential to set the worth for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.

By following this approach, we determine the range values and the domain for the function. Once we have 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 similar characteristics. When the base of an exponential function is greater than 1, the graph would have the following 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 flat and ongoing

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

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

In situations where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following characteristics:

  • The graph passes the point (0,1)

  • The range is more 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 level

  • The graph is constant


There are a few basic rules to recall when working with exponential functions.

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

For instance, if we need 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 the same base, subtract the exponents.

For instance, if we need to divide two exponential functions that posses a base of 3, we can compose 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 grow 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 equal to 1.

For instance, 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 instance, 0^x = 0 no matter what the value of x is.


Exponential functions are usually leveraged to denote exponential growth. As the variable rises, the value of the function increases faster and faster.

Example 1

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

At the end of the second hour, we will have 4x 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 total sum of bacteria at time t and t is measured hourly.

Example 2

Similarly, exponential functions can portray exponential decay. If we have a dangerous 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.

After the second hour, we will have one-fourth as much substance (1/2 x 1/2).

After hour three, we will have 1/8 as much material (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 measured in hours.

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

As a matter of 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 denoted by the variable whereas the base stays fixed. This indicates that any exponential growth or decline where the base changes is not an exponential function.

For example, in the case of compound interest, the interest rate remains the same while the base changes in ordinary time periods.


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

Let us check out this example.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As you can see, the worth of y increase very rapidly as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:

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

Example 2

Chart the following exponential function:

y = 1/2^x

First, let's make a table of values.

As shown, 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 draw the x-values and y-values on a coordinate plane, it would 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 written as f(ax)/dx = ax. All derivatives of exponential functions present special features where 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 common form of an exponential series is:


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