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 specific base. For instance, let us suppose a country's population doubles annually. 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.

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

What is the equation for an Exponential Function?

The general equation for an exponential function is f(x) = b^x, where:

b is the base, and x is the exponent or power.
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 does not equal 1, x will be a real number.

How do you plot Exponential Functions?

To graph an exponential function, we must discover the dots where the function crosses the axes. These are referred to as the x and y-intercepts.

Since 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 value for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.

In following this technique, we achieve the range values and the domain 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 similar properties. When the base of an exponential function is greater than 1, the graph is going to have the below characteristics:

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 constant
As x nears negative infinity, the graph is asymptomatic regarding the x-axis
As x nears positive infinity, the graph grows without bound.

In cases where the bases are fractions or decimals in the middle of 0 and 1, an exponential function exhibits the following attributes:

The graph crosses the point (0,1)
The range is greater than 0
The domain is all real numbers
The graph is descending
The graph is a curved line
As x approaches 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 unending

Rules

There are a few vital rules to remember when working with exponential functions.

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

For instance, if we need 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 identical base, subtract the exponents.

For example, if we need to divide two exponential functions with 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 instance, if we have to raise an exponential function with a base of 4 to the third power, then we can write 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 regardless of what the worth of x is.

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

For instance, 0^x = 0 regardless of what the value of x is.

Examples

Exponential functions are generally utilized to denote exponential growth. As the variable grows, the value of the function rises faster and faster.

Example 1

Let's look at the example of the growth of bacteria. Let us suppose that we have a group of bacteria that multiples by two every hour, then at the close of the first hour, we will have double as many bacteria.

At the end of the second hour, we will have quadruple 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 represented 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 in hours.

Example 2

Also, exponential functions can portray exponential decay. Let’s say we had a dangerous material that decays at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.

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

At the end of the third hour, we will have one-eighth as much material (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 amount of material at time t and t is calculated in hours.

As you can see, both of these examples use a comparable pattern, which is the reason they can be represented using exponential functions.

As a matter of 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 while the base remains fixed. Therefore any exponential growth or decline where the base changes is not an exponential function.

For example, in the case of compound interest, the interest rate remains the same whilst the base changes in regular intervals of time.

Solution

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

Let's review the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As you can see, the values of y increase very rapidly as x grows. 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 ,getting steeper as it persists.

Example 2

Plot the following exponential function:

y = 1/2^x

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

As you can see, the values of y decrease very rapidly as x increases. The reason is because 1/2 is less than 1.

Let’s say we were to chart the x-values and y-values on a coordinate plane, it would look like what you see below:

The above is a decay function. As shown, the graph is a curved line that gets lower from right to left and gets flatter as it goes.

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 display 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 terminology are the powers of an independent variable number. The general form of an exponential series is:

Source

Grade Potential is Able to Help You Learn Exponential Functions

If you're battling to comprehend exponential functions, or simply require a little extra support with math overall, consider partnering with a tutor. At Grade Potential, our Bakersfield math tutors are experts in their field and can provide you with the individualized consideration you need to succeed.

Call us at (661) 724-8887 or contact us today to find out more about the ways in which we can assist you in reaching your academic potential.

Back to Blog