June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function calculates an exponential decrease or increase in a specific base. For example, let us assume a country's population doubles yearly. This population growth can be depicted as an exponential function.

Exponential functions have multiple real-world use cases. Expressed mathematically, an exponential function is shown as f(x) = b^x.

Today we will review the essentials of an exponential function in conjunction with relevant 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 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 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 must find the spots where the function crosses the axes. This is referred to 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, 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 get the domain and the range values for the function. Once we have 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 will have the below characteristics:

The line passes the point (0,1)
The domain is all positive real numbers
The range is more than 0
The graph is a curved line
The graph is increasing
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 increases without bound.

In events where the bases are fractions or decimals in the middle of 0 and 1, an exponential function presents with the following properties:

The graph crosses the point (0,1)
The range is more than 0
The domain is entirely 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 smooth
The graph is constant

Rules

There are a few vital rules to bear in mind when working with exponential functions.

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

For instance, if we need to multiply two exponential functions with a base of 2, then we can note 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 compose 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, we are able to write it as (4^x)^3 = 4^(3x).

Rule 4: An exponential function with a base of 1 is consistently equal 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 equal to 0.

For instance, 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 increases faster and faster.

Example 1

Let's look at the example of the growth of bacteria. Let’s say we have a group of bacteria that duplicates every hour, then at the end of hour one, we will have 2 times as many bacteria.

At the end of the second hour, we will have quadruple 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 utilizing 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

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

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

After hour three, we will have 1/8 as much material (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 volume of substance at time t and t is calculated in hours.

As shown, both of these samples use a similar pattern, which is why 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 represented by the variable whereas the base stays constant. Therefore any exponential growth or decomposition where the base varies is not an exponential function.

For instance, in the case of compound interest, the interest rate continues to be the same whereas the base changes in regular time periods.

Solution

An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we need to enter different values for x and measure the equivalent values for y.

Let's look at 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 values of y increase very quickly as x grows. 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 ,getting steeper as it continues.

Example 2

Draw the following exponential function:

y = 1/2^x

First, let's create a table of values.

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

If we were to draw the x-values and y-values on a coordinate plane, it would look like this:

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 goes.

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 unique features 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 terminology are the powers of an independent variable figure. The common form of an exponential series is:

Source

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