June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

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

Exponential functions have many real-life applications. Expressed mathematically, an exponential function is displayed as f(x) = b^x.

Here we will review the fundamentals of an exponential function in conjunction with relevant examples.

What is the formula for an Exponential Function?

The general formula 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 example, 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 the event where b is higher than 0 and not equal to 1, x will be a real number.

How do you plot Exponential Functions?

To graph an exponential function, we must find the spots 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 discover the y-coordinates, its essential to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

According to this method, we achieve the domain and the range values for the function. After having the values, we need to plot them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share comparable characteristics. When the base of an exponential function is more than 1, the graph would 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 increasing
The graph is level and constant
As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
As x advances toward positive infinity, the graph rises 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 larger than 0
The domain is all real numbers
The graph is descending
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 flat
The graph is unending

Rules

There are a few basic rules to remember when engaging 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 an identical base, deduct the exponents.

For example, if we need to divide two exponential functions that have 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 increase 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 always equivalent 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 identical to 0.

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

Examples

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

Example 1

Let's look at the example of the growth of bacteria. If we have a group of bacteria that duplicates hourly, then at the close of the first hour, we will have twice 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 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 amount of bacteria at time t and t is measured in hours.

Example 2

Also, exponential functions can illustrate exponential decay. If we have a dangerous substance that degenerates at a rate of half its amount every hour, then at the end of the first hour, we will have half as much material.

At the end of hour two, we will have 1/4 as much material (1/2 x 1/2).

After three hours, 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 quantity of substance at time t and t is calculated in hours.

As shown, both of these illustrations use a comparable pattern, which is why they can be represented using exponential functions.

As a matter of fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is depicted by the variable while the base remains the same. This means that any exponential growth or decomposition where the base is different is not an exponential function.

For instance, in the scenario of compound interest, the interest rate remains the same whereas the base varies in normal amounts of time.

Solution

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

Let's review this example.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As you can see, the values of y increase very rapidly as x grows. 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 begin, let's create a table of values.

As you can see, the values of y decrease very quickly as x rises. This is because 1/2 is less than 1.

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

This is a decay function. As shown, the graph is a curved line that decreases 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 display special features whereby 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 terminology are the powers of an independent variable number. The common form of an exponential series is:

Source

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