June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function calculates an exponential decrease or increase in a certain base. Take this, for example, let's say a country's population doubles yearly. This population growth can be portrayed as an exponential function.

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

Here we discuss the basics of an exponential function along with relevant examples.

What is the formula 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 a constant, and x varies

For instance, 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 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 dots 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 locate the y-coordinates, we need to set the worth for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2

In following this technique, we determine the range values and the domain for the function. After having 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 properties. When the base of an exponential function is more than 1, the graph is going to have the below properties:

  • 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 continuous

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

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

In events where the bases are fractions or decimals between 0 and 1, an exponential function presents with the following qualities:

  • The graph passes the point (0,1)

  • The range is greater than 0

  • The domain is entirely real numbers

  • The graph is decreasing

  • The graph is a curved line

  • As x advances toward positive infinity, the line within graph is asymptotic to the x-axis.

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

  • The graph is level

  • The graph is continuous


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

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

For instance, if we need to multiply two exponential functions that posses 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 example, if we have to divide two exponential functions that posses a base of 3, we can write 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 increase an exponential function with a base of 4 to the third power, we are able to note it as (4^x)^3 = 4^(3x).

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

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

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

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


Exponential functions are generally utilized to signify 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 group of bacteria that doubles every hour, then at the end of the first hour, we will have 2 times 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 the third hour, we will have 8x as many bacteria (2 x 2 x 2).

This rate of growth can be represented using 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. If we have a dangerous 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.

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

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

This can be displayed 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 assessed in hours.

As demonstrated, both of these illustrations follow a comparable pattern, which is the reason they are able to be shown using exponential functions.

In 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 while the base remains constant. Therefore any exponential growth or decomposition where the base is different is not an exponential function.

For example, in the scenario of compound interest, the interest rate remains the same whereas the base varies in ordinary time periods.


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 asses the equivalent values for y.

Let's 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 demonstrated, the worth of y grow very rapidly as x increases. 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 goes up from left to right and gets steeper as it goes.

Example 2

Draw the following exponential function:

y = 1/2^x

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

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

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

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 special properties by which 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 common form of an exponential series is:


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