Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or increase in a certain base. Take this, for example, let's say a country's population doubles every year. This population growth can be represented 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 discuss the essentials of an exponential function along with important examples.
What is the equation for an Exponential Function?
The general 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 varies
As an illustration, 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 a situation where b is larger 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 need to locate the dots where the function intersects the axes. These are called the x and y-intercepts.
Since 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, its essential to set the value for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
By following this approach, we get the range values and the domain for the function. Once we determine the rate, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share similar characteristics. When the base of an exponential function is larger than 1, the graph would have the following properties:
-
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 on an incline
-
The graph is flat and ongoing
-
As x approaches negative infinity, the graph is asymptomatic concerning the x-axis
-
As x advances toward positive infinity, the graph grows without bound.
In situations where the bases are fractions or decimals between 0 and 1, an exponential function presents with the following attributes:
-
The graph passes the point (0,1)
-
The range is larger than 0
-
The domain is entirely real numbers
-
The graph is decreasing
-
The graph is a curved line
-
As x nears 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 level
-
The graph is constant
Rules
There are a few basic rules to recall when engaging with exponential functions.
Rule 1: Multiply exponential functions with an equivalent base, add the exponents.
For instance, 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 an identical base, subtract the exponents.
For example, if we need to divide two exponential functions that posses a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To increase an exponential function to a power, multiply the exponents.
For instance, if we have to grow an exponential function with a base of 4 to the third power, we are able to 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 example, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For example, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are commonly utilized to denote exponential growth. As the variable increases, the value of the function rises at a ever-increasing pace.
Example 1
Let’s examine the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that doubles every hour, then at the end of hour one, we will have 2 times as many bacteria.
At the end of hour two, we will have 4x as many bacteria (2 x 2).
At the end of hour three, we will have 8x 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 in hours.
Example 2
Also, exponential functions can illustrate 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 hour one, we will have half as much substance.
At the end of two hours, we will have one-fourth as much material (1/2 x 1/2).
After three hours, we will have one-eighth as much material (1/2 x 1/2 x 1/2).
This can be shown 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 calculated in hours.
As you can see, both of these illustrations pursue a similar pattern, which is why they are able to be represented 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 depicted by the variable whereas the base stays fixed. This means that any exponential growth or decomposition where the base changes is not an exponential function.
For instance, in the case of compound interest, the interest rate remains the same whereas the base changes 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 must input different values for x and measure the matching values for y.
Let's look at the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As you can see, the values of y increase very rapidly as x rises. Imagine 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 goes up from left to right and gets steeper as it continues.
Example 2
Plot the following exponential function:
y = 1/2^x
To begin, let's make a table of values.
As shown, the values of y decrease very rapidly as x surges. The reason 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 gets lower from right to left and gets flatter as it continues.
The Derivative of Exponential Functions
The derivative of an exponential function f(x) = a^x can be shown as f(ax)/dx = ax. All derivatives of exponential functions exhibit special characteristics 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:
Grade Potential is Able to Help You Succeed at Exponential Functions
If you're battling to understand exponential functions, or just need a little extra help with math overall, consider partnering with a tutor. At Grade Potential, our Akron math tutors are experts in their subjects and can provide you with the one-on-one support you need to triumph.
Call us at (330) 632-3388 or contact us now to learn more about the ways in which we can assist you in reaching your academic potential.
