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 certain base. For instance, let us suppose a country's population doubles yearly. This population growth can be depicted as an exponential function.

Exponential functions have numerous real-life uses. Expressed mathematically, an exponential function is written as f(x) = b^x.

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

What is the formula 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 the event where b is larger than 0 and not equal to 1, x will be a real number.

How do you graph Exponential Functions?

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

As the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.

To find 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 determine the range values and the domain for the function. Once we have the worth, we need to graph them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

All exponential functions share similar qualities. When the base of an exponential function is larger than 1, the graph would have the below qualities:

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 approaches negative infinity, the graph is asymptomatic regarding the x-axis
As x advances toward positive infinity, the graph rises without bound.

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

The graph crosses 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 in the 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 some basic rules to recall when working with exponential functions.

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

For example, if we need to multiply two exponential functions that have 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 identical base, subtract the exponents.

For instance, if we have to divide two exponential functions that have a base of 3, we can note it as 3^x / 3^y = 3^(x-y).

Rule 3: To grow 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 with a base of 1 is forever equal to 1.

For instance, 1^x = 1 no matter 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 used to indicate exponential growth. As the variable grows, the value of the function grows at a ever-increasing pace.

Example 1

Let’s examine the example of the growing of bacteria. If we have a culture of bacteria that multiples by two hourly, then at the end of the first hour, we will have double as many bacteria.

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

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

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

f(t) = 2^t

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

Example 2

Similarly, exponential functions can represent exponential decay. If we have a dangerous substance that decays at a rate of half its quantity every hour, then at the end of the first hour, we will have half as much substance.

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

After the third hour, we will have one-eighth 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 amount of material at time t and t is assessed in hours.

As shown, both of these samples use a similar pattern, which is the reason they can be shown using exponential functions.

In fact, any rate of change can be demonstrated using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable whereas the base continues to be fixed. This indicates that any exponential growth or decomposition where the base varies 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 amounts of time.

Solution

An exponential function is able to be graphed utilizing a table of values. To get the graph of an exponential function, we must input different values for x and then calculate the equivalent values for y.

Let us look at this example.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As demonstrated, the rates of y grow very quickly as x grows. If we were to draw 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 continues.

Example 2

Draw the following exponential function:

y = 1/2^x

To start, let's draw up a table of values.

As you can see, the values of y decrease very rapidly as x rises. This 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 what you see below:

The above is a decay function. As shown, the graph is a curved line that gets lower 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 special features whereby 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 figure. The common form of an exponential series is:

Source

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