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 specific base. Take this, for example, let us assume a country's population doubles annually. This population growth can be portrayed in the form of an exponential function.

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

Today we will review the basics of an exponential function coupled with relevant examples.

What’s the equation for an Exponential Function?

The common 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 is a variable

As an illustration, 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 cases where b is greater than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To chart an exponential function, we need to locate the spots where the function intersects the axes. This is referred to as the x and y-intercepts.

As 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, its essential to set the worth for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

According to this approach, we determine the range values and the domain for the function. After having the rate, we need to chart 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 will 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 on an incline
The graph is level and continuous
As x nears negative infinity, the graph is asymptomatic towards the x-axis
As x nears positive infinity, the graph grows without bound.

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

The graph intersects the point (0,1)
The range is more than 0
The domain is entirely real numbers
The graph is decreasing
The graph is a curved line
As x nears positive infinity, the line in the graph is asymptotic to the x-axis.
As x gets closer to negative infinity, the line approaches without bound
The graph is level
The graph is continuous

Rules

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

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

For instance, if we have 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 raise an exponential function to a power, multiply the exponents.

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

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

For instance, 1^x = 1 no matter what the rate of x is.

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

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

Examples

Exponential functions are generally leveraged to denote exponential growth. As the variable increases, the value of the function grows at a ever-increasing pace.

Example 1

Let’s examine the example of the growth of bacteria. Let’s say we have a culture of bacteria that doubles each hour, then at the end of the first hour, we will have twice as many bacteria.

At the end of the second hour, we will have 4 times as many bacteria (2 x 2).

At the end of hour three, we will have 8x as many bacteria (2 x 2 x 2).

This rate of growth can be represented an exponential function as follows:

f(t) = 2^t

where f(t) is the number of bacteria at time t and t is measured hourly.

Example 2

Also, exponential functions can illustrate exponential decay. Let’s say we had a radioactive material that degenerates at a rate of half its volume every hour, then at the end of hour one, we will have half as much substance.

At the end of the second hour, we will have one-fourth as much material (1/2 x 1/2).

After three hours, we will have an eighth as much material (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 amount of substance at time t and t is assessed in hours.

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

In fact, any rate of change can be demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is represented by the variable while the base continues to be the same. Therefore any exponential growth or decline where the base is different is not an exponential function.

For example, in the matter of compound interest, the interest rate continues to be the same whilst the base varies in regular 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 plug in different values for x and measure the matching values for y.

Let us review the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As demonstrated, the rates of y grow very rapidly as x rises. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:

As you can see, the graph is a curved line that rises from left to right and gets steeper as it persists.

Example 2

Graph the following exponential function:

y = 1/2^x

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

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

Let’s say we were to plot the x-values and y-values on a coordinate plane, it would look like this:

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

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be displayed as f(ax)/dx = ax. All derivatives of exponential functions display special features where 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

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