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 particular base. For instance, let us assume a country's population doubles every year. This population growth can be portrayed as an exponential function.

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

Today we will review the essentials of an exponential function along with important examples.

What is 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 a constant, and x is a variable

For instance, 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 does not equal 1, x will be a real number.

How do you plot Exponential Functions?

To chart an exponential function, we must locate the dots where the function crosses the axes. This is called the x and y-intercepts.

Since the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.

To locate the y-coordinates, one must 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 technique, we achieve the domain and the range values for the function. Once we determine the worth, we need to chart them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share similar properties. When the base of an exponential function is larger than 1, the graph will 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 rising
The graph is flat and continuous
As x approaches negative infinity, the graph is asymptomatic regarding the x-axis
As x nears positive infinity, the graph grows without bound.

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

The graph passes the point (0,1)
The range is greater than 0
The domain is all real numbers
The graph is descending
The graph is a curved line
As x approaches positive infinity, the line in the graph is asymptotic to the x-axis.
As x approaches negative infinity, the line approaches without bound
The graph is level
The graph is continuous

Rules

There are a few essential rules to remember when working with exponential functions.

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

For example, if we have to multiply two exponential functions that posses a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with the same base, subtract the exponents.

For instance, if we have 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 example, if we have to grow 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 that has a base of 1 is always equivalent to 1.

For example, 1^x = 1 regardless of what the value of x is.

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

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

Examples

Exponential functions are generally utilized to indicate exponential growth. As the variable grows, the value of the function rises quicker and quicker.

Example 1

Let's look at the example of the growth of bacteria. Let us suppose that we have a culture of bacteria that duplicates each hour, then at the close of hour one, we will have twice as many bacteria.

At the end of the second hour, 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 displayed utilizing an exponential function as follows:

f(t) = 2^t

where f(t) is the total sum of bacteria at time t and t is measured in hours.

Example 2

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

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

At the end of hour three, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).

This can be displayed using an exponential equation as follows:

f(t) = 1/2^t

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

As you can see, both of these examples use a similar pattern, which is the reason they are able to be represented using exponential functions.

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

For instance, in the scenario of compound interest, the interest rate remains the same whilst the base changes in ordinary time periods.

Solution

An exponential function can be graphed using a table of values. To get the graph of an exponential function, we must input different values for x and calculate the matching values for y.

Let's review this example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As shown, the rates of y grow very fast as x increases. 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 rises from left to right ,getting steeper as it persists.

Example 2

Plot 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 rapidly as x increases. This is because 1/2 is less than 1.

If we were to plot 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 you can see, the graph is a curved line that descends 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions display special properties 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 terminology are the powers of an independent variable figure. The general form of an exponential series is:

Source

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