June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function calculates an exponential decrease or rise in a certain base. Take this, for example, let's say a country's population doubles annually. This population growth can be depicted in the form of an exponential function.

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

Here we will learn the basics of an exponential function coupled with relevant examples.

What’s the formula 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 fixed, 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 not equal to 1, x will be a real number.

How do you plot Exponential Functions?

To plot an exponential function, we have to find the points where the function intersects the axes. These are known as the x and y-intercepts.

As the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.

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

In following this technique, we get the domain and the range values for the function. Once we determine the rate, we need to graph 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 would have the following qualities:

The line passes the point (0,1)
The domain is all positive real numbers
The range is greater than 0
The graph is a curved line
The graph is rising
The graph is flat and continuous
As x nears negative infinity, the graph is asymptomatic towards the x-axis
As x approaches positive infinity, the graph increases without bound.

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

The graph intersects the point (0,1)
The range is larger than 0
The domain is all real numbers
The graph is declining
The graph is a curved line
As x approaches 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 unending

Rules

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

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

For example, 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 the same base, deduct the exponents.

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

Rule 4: An exponential function with a base of 1 is always equivalent to 1.

For example, 1^x = 1 no matter 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.

Examples

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

Example 1

Let's look at the example of the growth of bacteria. Let’s say we have a cluster of bacteria that duplicates each hour, then at the close of the first hour, 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 hour three, we will have 8 times as many bacteria (2 x 2 x 2).

This rate of growth can be displayed 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

Similarly, exponential functions can represent exponential decay. If we have a radioactive material that degenerates at a rate of half its volume every hour, then at the end of one hour, we will have half as much material.

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

After 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 below:

f(t) = 1/2^t

where f(t) is the amount of material at time t and t is measured in hours.

As shown, both of these illustrations follow a similar pattern, which is the reason they can be depicted 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 depicted by the variable whereas the base remains constant. Therefore any exponential growth or decline where the base varies is not an exponential function.

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

Solution

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

Let us check out this example.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As demonstrated, the worth of y increase very rapidly as x rises. If we were to draw this exponential function graph on a coordinate plane, it would look like this:

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

Example 2

Graph the following exponential function:

y = 1/2^x

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

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

Let’s say we were to graph the x-values and y-values on a coordinate plane, it is going to look like the following:

This is a decay function. As you can see, the graph is a curved line that descends 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 shown as f(ax)/dx = ax. All derivatives of exponential functions present particular characteristics 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:

Source

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