Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a specific base. Take this, for example, let's say a country's population doubles yearly. This population growth can be portrayed in the form of an exponential function.
Exponential functions have numerous real-life use cases. Expressed mathematically, an exponential function is written as f(x) = b^x.
Today we discuss the basics of an exponential function coupled with appropriate examples.
What is the formula for an Exponential Function?
The generic equation for an exponential function is f(x) = b^x, where:
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b is the base, and x is the exponent or power.
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b is a constant, 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 a situation where b is higher than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To graph an exponential function, we have to find the dots where the function intersects the axes. This is known as 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 take the value of b = 2.
To find the y-coordinates, one must to set the rate for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
According to this method, we achieve the range values and the domain for the function. Once we determine the worth, 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 qualities. When the base of an exponential function is greater than 1, the graph will have the below properties:
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The line passes the point (0,1)
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The domain is all positive real numbers
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The range is more than 0
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The graph is a curved line
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The graph is increasing
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The graph is flat and constant
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As x advances toward negative infinity, the graph is asymptomatic concerning the x-axis
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As x nears positive infinity, the graph rises without bound.
In instances where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following properties:
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The graph intersects the point (0,1)
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The range is greater than 0
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The domain is entirely real numbers
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The graph is descending
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The graph is a curved line
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As x approaches positive infinity, the line in the graph is asymptotic to the x-axis.
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As x gets closer to negative infinity, the line approaches without bound
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The graph is flat
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The graph is constant
Rules
There are a few basic rules to bear in mind when working with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, if we need to multiply two exponential functions with a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, subtract the exponents.
For example, 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 increase 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, we are able to write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with 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 equivalent to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are usually used to denote exponential growth. As the variable rises, the value of the function grows faster and faster.
Example 1
Let’s examine the example of the growing of bacteria. If we have a culture of bacteria that doubles each hour, then at the close of hour one, we will have 2 times as many bacteria.
At the end of the second hour, we will have 4x 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 portrayed 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
Also, exponential functions can illustrate exponential decay. If we have a radioactive material that degenerates at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.
After two hours, we will have a quarter as much substance (1/2 x 1/2).
At the end of hour three, 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 below:
f(t) = 1/2^t
where f(t) is the volume of material at time t and t is calculated in hours.
As you can see, both of these examples use a similar pattern, which is why they are able to be shown using exponential functions.
In fact, any rate of change can be indicated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base stays fixed. This means that any exponential growth or decline where the base varies 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 intervals of time.
Solution
An exponential function is able to 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 equivalent values for y.
Let's check out 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 worth of y increase very fast 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 ,getting steeper as it goes.
Example 2
Graph 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 rapidly as x surges. This is because 1/2 is less than 1.
If we were to draw the x-values and y-values on a coordinate plane, it is going to look like this:
The above is a decay function. As shown, the graph is a curved line that gets lower 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions display particular properties 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 digit. The general form of an exponential series is:
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