June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or increase in a certain base. Take this, for example, let's say a country's population doubles every year. This population growth can be depicted in the form of an exponential function.

Exponential functions have multiple real-world uses. Mathematically speaking, an exponential function is written as f(x) = b^x.

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

What is the equation for an Exponential Function?

The generic 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 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 cases where b is higher 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 need to locate the points where the function crosses the axes. These are referred to as the x and y-intercepts.

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

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

By following this method, we get the domain and the range values for the function. Once we have the worth, we need to chart 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 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 continuous

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

  • As x nears positive infinity, the graph grows without bound.

In situations where the bases are fractions or decimals within 0 and 1, an exponential function exhibits the following properties:

  • The graph crosses the point (0,1)

  • The range is greater than 0

  • The domain is all 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 smooth

  • The graph is constant


There are some basic rules to bear in mind when engaging 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 have 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, subtract the exponents.

For instance, if we have to divide two exponential functions with 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 example, 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 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 no matter what the value of x is.


Exponential functions are generally utilized to denote exponential growth. As the variable increases, the value of the function increases faster and faster.

Example 1

Let's look at the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that doubles every hour, then at the end 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 portrayed 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 hourly.

Example 2

Similarly, exponential functions can represent exponential decay. Let’s say we had a radioactive material that decomposes at a rate of half its amount every hour, then at the end of one hour, we will have half as much material.

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

At the end of the third hour, we will have 1/8 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 volume of substance at time t and t is assessed in hours.

As shown, both of these samples pursue a similar pattern, which is why they can be represented using exponential functions.

As a matter of fact, any rate of change can be denoted using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base remains fixed. This indicates that any exponential growth or decline where the base changes is not an exponential function.

For instance, in the matter of compound interest, the interest rate continues to be the same while the base varies in normal amounts of time.


An exponential function can be graphed utilizing 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 look at 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 worth of y grow very fast as x grows. If we were to draw this exponential function graph on a coordinate plane, it would look like the following:

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

Example 2

Chart the following exponential function:

y = 1/2^x

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

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

Let’s say we were to chart the x-values and y-values on a coordinate plane, it is going to look like the following:

This 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 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 present special characteristics 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 digit. The common form of an exponential series is:


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