Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a particular base. Take this, for example, let us suppose a country's population doubles annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have numerous real-life use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Today we will review the basics of an exponential function along with appropriate examples.
What is the equation for an Exponential Function?
The generic equation 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
As an illustration, 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 greater than 0 and not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To chart an exponential function, we need to locate the spots where the function crosses the axes. These are known as the x and y-intercepts.
As the exponential function has a constant, we need to set the value for it. Let's focus on the value of b = 2.
To locate the y-coordinates, we need to set the rate for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
In following this approach, we get the domain and the range values for the function. After having the rate, 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 passes 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 flat and continuous
-
As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
-
As x approaches positive infinity, the graph grows without bound.
In cases where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties:
-
The graph intersects the point (0,1)
-
The range is larger than 0
-
The domain is entirely 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 gets closer to negative infinity, the line approaches without bound
-
The graph is level
-
The graph is continuous
Rules
There are a few vital rules to remember when dealing with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For example, if we need to multiply two exponential functions that posses a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an identical base, subtract the exponents.
For example, if we need 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 raise an exponential function to a power, multiply the exponents.
For example, if we have to raise 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 that has a base of 1 is forever equivalent 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 used to denote exponential growth. As the variable rises, the value of the function grows quicker and quicker.
Example 1
Let’s observe the example of the growing of bacteria. If we have a cluster of bacteria that multiples by two each hour, then at the end of the first hour, we will have twice as many bacteria.
At the end of the second hour, 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 displayed utilizing an exponential function as follows:
f(t) = 2^t
where f(t) is the amount of bacteria at time t and t is measured in hours.
Example 2
Also, exponential functions can represent exponential decay. Let’s say we had a dangerous substance that decomposes at a rate of half its quantity every hour, then at the end of one hour, we will have half as much material.
After hour two, we will have a quarter as much material (1/2 x 1/2).
At the end of three hours, we will have an eighth as much material (1/2 x 1/2 x 1/2).
This can be shown 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 pursue a comparable pattern, which is the reason they are able to 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 while the base continues to be constant. This indicates that any exponential growth or decomposition where the base varies is not an exponential function.
For example, in the scenario of compound interest, the interest rate stays the same whereas the base varies in ordinary time periods.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we need to input different values for x and asses the matching values for y.
Let us review 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 worth of y increase very quickly as x grows. 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
To start, let's make a table of values.
As you can see, the values of y decrease very quickly as x rises. This is because 1/2 is less than 1.
If we were to chart the x-values and y-values on a coordinate plane, it would look like the following:
The above is a decay function. As shown, the graph is a curved line that decreases 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 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 terms are the powers of an independent variable figure. The common form of an exponential series is:
Grade Potential is Able to Help You Learn Exponential Functions
If you're fighting to comprehend exponential functions, or just require some extra support with math as a whole, consider working with a tutor. At Grade Potential, our Pembroke math tutors are experts in their subjects and can provide you with the one-on-one support you need to thrive.
Call us at (954) 448-7531 or contact us now to learn more about the ways in which we can assist you in reaching your academic potential.
