June 03, 2022

Exponential Functions - Formula, Properties, Graph, Rules

What is an Exponential Function?

An exponential function measures an exponential decrease or increase in a particular base. For example, let us suppose a country's population doubles every year. This population growth can be represented in the form of an exponential function.

Exponential functions have multiple real-world applications. Mathematically speaking, an exponential function is displayed as f(x) = b^x.

Here we will review the essentials of an exponential function in conjunction with relevant examples.

What is the equation for an Exponential Function?

The general formula for an exponential function is f(x) = b^x, where:

  1. b is the base, and x is the exponent or power.

  2. b is fixed, and x is a variable

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 cases where b is higher than 0 and unequal to 1, x will be a real number.

How do you plot Exponential Functions?

To plot an exponential function, we need to discover the points where the function crosses the axes. This is called the x and y-intercepts.

Considering the fact that 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, we need to set the value for x. For example, for x = 2, y will be 4, for x = 1, y will be 2

According to this method, we determine the range values and the domain for the function. Once we determine the values, we need to draw them on the x-axis and the y-axis.

What are the properties of Exponential Functions?

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

  • The line crosses 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 ongoing

  • 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 instances where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following attributes:

  • 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 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 flat

  • The graph is continuous

Rules

There are several basic rules to recall when engaging 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 posses a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).

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

For instance, if we need 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 example, if we have to increase an exponential function with a base of 4 to the third power, then we can compose it as (4^x)^3 = 4^(3x).

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

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

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

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

Examples

Exponential functions are generally used to indicate exponential growth. As the variable rises, the value of the function grows 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 hour one, we will have double as many bacteria.

At the end of hour two, we will have 4 times 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 represented 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 hourly.

Example 2

Moreover, exponential functions can portray exponential decay. If we have a radioactive material that decays at a rate of half its volume every hour, then at the end of hour one, we will have half as much substance.

After hour two, we will have a quarter as much substance (1/2 x 1/2).

After the third hour, we will have 1/8 as much material (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 amount of substance at time t and t is measured in hours.

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

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

For example, in the scenario of compound interest, the interest rate continues to be the same whilst the base changes in ordinary amounts of time.

Solution

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

Let us look at the example below.

Example 1

Graph the this exponential function formula:

y = 3^x

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

As shown, the values of y increase very rapidly as x increases. If we were to plot this exponential function graph on a coordinate plane, it would look like the following:

As shown, the graph is a curved line that rises from left to right ,getting steeper as it goes.

Example 2

Plot the following exponential function:

y = 1/2^x

First, let's draw up a table of values.

As shown, the values of y decrease very swiftly as x increases. This is because 1/2 is less than 1.

Let’s say we were to draw 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 you can see, the graph is a curved line that decreases from right to left and gets smoother as it proceeds.

The Derivative of Exponential Functions

The derivative of an exponential function f(x) = a^x can be written as f(ax)/dx = ax. All derivatives of exponential functions display unique 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 digit. The general form of an exponential series is:

Source

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