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

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

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

What’s the formula 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, 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 higher than 0 and does not equal 1, x will be a real number.

How do you graph Exponential Functions?

To chart an exponential function, we must find the points where the function intersects the axes. This is referred to as the x and y-intercepts.

As 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 discover 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

By following this technique, we get the range values and the domain 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 comparable characteristics. When the base of an exponential function is larger than 1, the graph is going to have the below characteristics:

The line passes 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 on an incline
The graph is smooth and continuous
As x nears negative infinity, the graph is asymptomatic towards the x-axis
As x advances toward positive infinity, the graph increases without bound.

In situations where the bases are fractions or decimals within 0 and 1, an exponential function displays 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 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 gets closer to negative infinity, the line approaches without bound
The graph is flat
The graph is continuous

Rules

There are several vital rules to remember when engaging 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 compose it as 2^x * 2^y = 2^(x+y).

Rule 2: To divide exponential functions with an identical base, deduct the exponents.

For example, if we need to divide two exponential functions with a base of 3, we can note 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 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 that has a base of 1 is always equal to 1.

For example, 1^x = 1 regardless of what the worth 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 indicate exponential growth. As the variable increases, 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 us suppose that we have a culture of bacteria that duplicates hourly, then at the close of the first hour, 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 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 total sum of bacteria at time t and t is measured hourly.

Example 2

Similarly, exponential functions can represent 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 substance.

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

After 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 quantity of material at time t and t is calculated in hours.

As shown, both of these examples use a comparable pattern, which is why they can be depicted using exponential functions.

As a matter of fact, any rate of change can be denoted using exponential functions. Recall that in exponential functions, the positive or the negative exponent is represented by the variable whereas the base stays fixed. Therefore any exponential growth or decline where the base is different is not an exponential function.

For example, in the scenario of compound interest, the interest rate remains the same whereas the base is static in regular intervals of time.

Solution

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

Let us look at this example.

Example 1

Graph the this exponential function formula:

y = 3^x

First, let's make a table of values.

As you can see, the rates of y grow very fast as x grows. If we were to draw this exponential function graph on a coordinate plane, it would look like the following:

As shown, the graph is a curved line that goes up from left to right and gets steeper as it continues.

Example 2

Chart the following exponential function:

y = 1/2^x

To begin, let's draw up 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 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 descends from right to left and gets flatter 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 present unique 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 terms are the powers of an independent variable figure. The general form of an exponential series is:

Source

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