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. Take this, for example, let's say a country's population doubles annually. This population growth can be represented as an exponential function.

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

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

What’s the formula for an Exponential Function?

The general 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

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 a situation where b is greater 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 have to discover the points where the function crosses the axes. These are called the x and y-intercepts.

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

To discover the y-coordinates, one must to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

By following this approach, we get the domain and the range values for the function. Once we determine 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 identical characteristics. When the base of an exponential function is greater than 1, the graph is going to have the below 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 rising

  • The graph is flat and continuous

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

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

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

  • The graph intersects the point (0,1)

  • The range is larger than 0

  • The domain is entirely real numbers

  • The graph is declining

  • 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 flat

  • The graph is constant


There are a few basic rules to bear in mind when engaging with exponential functions.

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

For example, if we have to multiply two exponential functions with 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 instance, if we have to divide two exponential functions with 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, 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 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 equal to 0.

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


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

Example 1

Let's look at the example of the growth of bacteria. Let us suppose that we have a cluster of bacteria that multiples by two each hour, then at the end of the first hour, 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 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

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

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

After hour three, we will have an eighth as much material (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 quantity of substance at time t and t is assessed in hours.

As you can see, both of these illustrations use a similar pattern, which is why they are able to be represented using exponential functions.

In 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 represented by the variable while the base continues to be fixed. Therefore any exponential growth or decline where the base is different is not an exponential function.

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


An exponential function can be graphed employing a table of values. To get the graph of an exponential function, we must enter different values for x and then calculate the corresponding values for y.

Let's review 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 rates of y increase very fast as x increases. Consider we were to draw this exponential function graph on a coordinate plane, it would look like this:

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

Example 2

Chart the following exponential function:

y = 1/2^x

To start, let's draw up a table of values.

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

If we were to draw the x-values and y-values on a coordinate plane, it would 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 flatter 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 present particular characteristics 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 expressions are the powers of an independent variable number. The general form of an exponential series is:


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