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 certain base. For example, let us assume a country's population doubles annually. This population growth can be represented as an exponential function.

Exponential functions have many real-world uses. In mathematical terms, an exponential function is shown as f(x) = b^x.

Today we will learn the basics of an exponential function coupled with important examples.

What is the formula for an Exponential Function?

The generic 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 cases where b is larger than 0 and not equal to 1, x will be a real number.

How do you graph Exponential Functions?

To plot an exponential function, we have to find the spots where the function crosses the axes. This is known 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 locate the y-coordinates, we need to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2

In following this technique, we achieve the range values and the domain for the function. Once we determine the rate, 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 qualities. When the base of an exponential function is greater than 1, the graph is going to have the following 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 increasing
The graph is level and continuous
As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
As x advances toward positive infinity, the graph increases without bound.

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

The graph crosses the point (0,1)
The range is more than 0
The domain is all 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 gets closer to negative infinity, the line approaches without bound
The graph is smooth
The graph is unending

Rules

There are several basic rules to remember when working with exponential functions.

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

For example, if we need to multiply two exponential functions with a base of 2, then we can compose 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 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 increase 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 consistently equivalent to 1.

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

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

For example, 0^x = 0 despite whatever 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 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 multiples by two hourly, then at the close of hour one, we will have twice as many bacteria.

At the end of hour two, 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 portrayed 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. Let’s say we had a radioactive substance that decomposes 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 two hours, we will have 1/4 as much substance (1/2 x 1/2).

After hour three, we will have an eighth as much substance (1/2 x 1/2 x 1/2).

This can be represented using an exponential equation as below:

f(t) = 1/2^t

where f(t) is the quantity of material at time t and t is assessed in hours.

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

In fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable while the base stays the same. This indicates that any exponential growth or decomposition where the base is different is not an exponential function.

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

Solution

An exponential function is able to be graphed utilizing a table of values. To get the graph of an exponential function, we have to plug in different values for x and measure 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 you can see, the rates of y grow very quickly as x grows. Imagine 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 goes.

Example 2

Graph the following exponential function:

y = 1/2^x

First, let's draw up a table of values.

As shown, the values of y decrease very rapidly as x increases. This is because 1/2 is less than 1.

If we were to graph the x-values and y-values on a coordinate plane, it would 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 smoother 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 display special features by which 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 terms are the powers of an independent variable number. The common form of an exponential series is:

Source

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