June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or increase in a particular base. For example, let us assume a country's population doubles yearly. This population growth can be depicted in the form of an exponential function.

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

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

What is 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 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 the event where b is larger than 0 and unequal to 1, x will be a real number.

How do you graph Exponential Functions?

To chart an exponential function, we need to locate the dots 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 focus on the value of b = 2.

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

In following this technique, we get the range values and the domain for the function. Once we have the rate, 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 qualities. When the base of an exponential function is larger than 1, the graph will have the following properties:

  • The line intersects 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 rising

  • The graph is level and constant

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

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

In instances where the bases are fractions or decimals in the middle of 0 and 1, an exponential function presents with the following properties:

  • The graph passes the point (0,1)

  • The range is larger than 0

  • The domain is entirely real numbers

  • The graph is descending

  • The graph is a curved line

  • As x nears 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 several basic 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 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 the same base, subtract 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 increase an exponential function to a power, multiply the exponents.

For instance, if we have to raise 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 forever equal to 1.

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

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

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


Exponential functions are usually utilized to denote exponential growth. As the variable rises, the value of the function increases 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 culture of bacteria that multiples by two hourly, then at the close of hour one, we will have twice as many bacteria.

At the end of the second hour, we will have quadruple 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 represented an exponential function as follows:

f(t) = 2^t

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

Example 2

Also, exponential functions can illustrate exponential decay. If we have 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 substance.

At the end of the second hour, we will have a quarter as much material (1/2 x 1/2).

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

This can be represented using an exponential equation as follows:

f(t) = 1/2^t

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

As shown, both of these examples follow a similar pattern, which is the reason they can be shown using exponential functions.

In fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base remains the same. Therefore any exponential growth or decomposition where the base varies is not an exponential function.

For instance, in the scenario of compound interest, the interest rate stays the same whilst 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 have to input different values for x and calculate the equivalent values for y.

Let us check out this example.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As shown, the worth of y grow very quickly as x rises. Imagine 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 persists.

Example 2

Plot 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 swiftly as x surges. This is because 1/2 is less than 1.

Let’s say 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 shown, the graph is a curved line that decreases from right to left and gets flatter 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 present particular features where 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 expressions are the powers of an independent variable figure. The common form of an exponential series is:


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