June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What’s an Exponential Function?

An exponential function measures an exponential decrease or increase in a particular base. For instance, let us assume a country's population doubles annually. This population growth can be depicted as an exponential function.

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

Here we discuss the fundamentals of an exponential function along with important examples.

What’s the equation 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 a constant, and x varies

For example, 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 a situation where b is greater than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To plot an exponential function, we have to discover the dots where the function intersects the axes. This is known as 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, one must to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

In following this method, we achieve the domain and the range values for the function. After having the values, 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 characteristics:

The line intersects 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 on an incline
The graph is smooth and continuous
As x nears 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 between 0 and 1, an exponential function exhibits the following properties:

The graph intersects the point (0,1)
The range is greater than 0
The domain is entirely real numbers
The graph is declining
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 level
The graph is unending

Rules

There are some basic rules to remember when dealing with exponential functions.

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

For instance, if we have to multiply two exponential functions that have 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 have to divide two exponential functions that posses a base of 3, we can write 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 write it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function that has a base of 1 is consistently equal to 1.

For example, 1^x = 1 no matter what the worth of x is.

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

For example, 0^x = 0 no matter what the value of x is.

Examples

Exponential functions are usually used to signify exponential growth. As the variable increases, the value of the function rises quicker and quicker.

Example 1

Let’s examine the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that multiples by two every hour, then at the end 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 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 number 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 amount every hour, then at the end of hour one, we will have half as much material.

After hour two, we will have 1/4 as much substance (1/2 x 1/2).

At the end of hour three, 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 calculated in hours.

As demonstrated, both of these samples follow a similar pattern, which is why they are able to be depicted using exponential functions.

In fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable while the base continues to be fixed. This means that any exponential growth or decay where the base changes is not an exponential function.

For instance, in the case of compound interest, the interest rate remains the same whilst the base varies in regular amounts 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 have to plug in different values for x and then asses the equivalent values for y.

Let's look at the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As you can see, the values of y grow very fast as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:

As shown, the graph is a curved line that rises 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 increases. The reason is because 1/2 is less than 1.

If we were to graph the x-values and y-values on a coordinate plane, it is going to 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 smoother 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 particular characteristics 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 terms are the powers of an independent variable number. The general form of an exponential series is:

Source

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