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. For instance, let's say a country's population doubles yearly. This population growth can be portrayed as an exponential function.

Exponential functions have numerous real-world 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 appropriate examples.

What’s the equation for an Exponential Function?

The generic 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 instance, 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 graph Exponential Functions?

To plot an exponential function, we have to discover the dots where the function intersects the axes. These are called the x and y-intercepts.

Since the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.

To discover the y-coordinates, its essential to set the rate for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

In following this technique, we get the range values and the domain for the function. Once we have the values, we need to chart them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share identical properties. When the base of an exponential function is greater than 1, the graph is going to have the below characteristics:

  • The line passes 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 increasing

  • The graph is smooth and constant

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

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

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

  • The graph crosses 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 nears 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 continuous


There are several vital rules to bear in mind when dealing with exponential functions.

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

For example, if we have to multiply two exponential functions with a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).

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

For instance, if we have to divide two exponential functions that have a base of 3, we can note 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 increase an exponential function with a base of 4 to the third power, then we can compose it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is forever equal to 1.

For instance, 1^x = 1 no matter what the value 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 generally leveraged to signify exponential growth. As the variable grows, the value of the function rises quicker and quicker.

Example 1

Let’s observe the example of the growing of bacteria. If we have a cluster of bacteria that multiples by two hourly, then at the end of hour one, we will have 2 times 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 portrayed 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

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

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

At the end of the third hour, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be shown using an exponential equation as below:

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 examples pursue a comparable pattern, which is why they can be represented using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base stays the same. Therefore any exponential growth or decline where the base is different is not an exponential function.

For instance, in the case of compound interest, the interest rate continues to be the same whilst the base is static in ordinary amounts of time.


An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we must enter different values for x and asses the matching values for y.

Let's look at this example.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As demonstrated, the values of y increase very rapidly as x rises. Consider we were to draw 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 ,getting steeper as it goes.

Example 2

Plot 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. This 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 this:

This is a decay function. As you can see, the graph is a curved line that descends from right to left and gets smoother as it continues.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions display special properties whereby 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 terminology are the powers of an independent variable digit. The general form of an exponential series is:


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