June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function measures an exponential decrease or rise in a particular base. For instance, let's say a country's population doubles every year. This population growth can be represented in the form of an exponential function.

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

Here we will review the essentials of an exponential function along with relevant 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 fixed, and x varies

For example, 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 cases 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 need to find the dots where the function intersects the axes. This is referred to as the x and y-intercepts.

Considering the fact that 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, its essential to set the worth for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.

According to this method, we achieve the range values and the domain for the function. Once we determine 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 would 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 flat and ongoing

  • As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis

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

In events where the bases are fractions or decimals within 0 and 1, an exponential function exhibits 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 within graph is asymptotic to the x-axis.

  • As x advances toward negative infinity, the line approaches without bound

  • The graph is flat

  • The graph is continuous


There are a few basic rules to remember 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 have a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).

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

For instance, if we have 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 example, if we have to grow 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 with a base of 1 is consistently equivalent to 1.

For instance, 1^x = 1 regardless of what the worth of x is.

Rule 5: An exponential function with a base of 0 is always equivalent to 0.

For example, 0^x = 0 despite whatever the value of x is.


Exponential functions are usually utilized to signify exponential growth. As the variable rises, the value of the function rises at a ever-increasing pace.

Example 1

Let's look at the example of the growing of bacteria. If we have a group of bacteria that duplicates each hour, then at the end of the first hour, 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 8x as many bacteria (2 x 2 x 2).

This rate of growth can be represented utilizing an exponential function as follows:

f(t) = 2^t

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

Example 2

Also, exponential functions can illustrate exponential decay. If we have a dangerous material 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 the second hour, we will have a quarter as much material (1/2 x 1/2).

After hour three, we will have 1/8 as much substance (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as follows:

f(t) = 1/2^t

where f(t) is the amount of substance at time t and t is calculated in hours.

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

In fact, any rate of change can be denoted using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable while the base remains constant. This indicates that any exponential growth or decay where the base varies is not an exponential function.

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


An exponential function is able to be graphed employing a table of values. To get the graph of an exponential function, we must plug in different values for x and measure the corresponding values for y.

Let us check out the following 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 rise very quickly as x rises. Imagine 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 persists.

Example 2

Draw the following exponential function:

y = 1/2^x

First, let's make a table of values.

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

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

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 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 exhibit unique features whereby 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 digit. The common form of an exponential series is:


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