Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or increase in a specific base. For example, let us suppose a country's population doubles annually. This population growth can be represented as an exponential function.
Exponential functions have many real-world applications. In mathematical terms, an exponential function is written as f(x) = b^x.
In this piece, we discuss the basics of an exponential function in conjunction with appropriate examples.
What is the equation for an Exponential Function?
The common formula 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 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 cases where b is greater than 0 and not equal to 1, x will be a real number.
How do you graph Exponential Functions?
To graph an exponential function, we have to locate the points where the function crosses the axes. These are called the x and y-intercepts.
As 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, we need to set the worth for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
In following this method, we get the domain and the range values for the function. After having the values, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share identical qualities. When the base of an exponential function is more than 1, the graph would have the below qualities:
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The line intersects the point (0,1)
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The domain is all positive real numbers
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The range is larger 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 level and constant
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As x nears negative infinity, the graph is asymptomatic towards the x-axis
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As x advances toward positive infinity, the graph increases without bound.
In cases where the bases are fractions or decimals in the middle of 0 and 1, an exponential function presents with the following attributes:
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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 all 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 advances toward negative infinity, the line approaches without bound
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The graph is flat
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The graph is unending
Rules
There are some basic rules to recall when working 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 write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an identical base, deduct 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 instance, if we have to grow an exponential function with a base of 4 to the third power, we are able to note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equivalent to 1.
For example, 1^x = 1 no matter what the rate of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are usually used to signify exponential growth. As the variable grows, the value of the function rises quicker and quicker.
Example 1
Let’s observe the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that duplicates 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 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 represented utilizing an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured in hours.
Example 2
Moreover, exponential functions can represent exponential decay. Let’s say we had a dangerous substance 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 two hours, we will have a quarter as much substance (1/2 x 1/2).
After the third hour, we will have an eighth 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 calculated in hours.
As shown, both of these samples follow a similar pattern, which is why they can be shown using exponential functions.
In fact, any rate of change can be demonstrated using exponential functions. Keep in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base remains the same. This means that any exponential growth or decline where the base changes is not an exponential function.
For example, in the case of compound interest, the interest rate remains the same while the base changes in normal intervals of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we must enter different values for x and then asses the corresponding values for y.
Let's look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As you can see, the values of y increase very fast as x grows. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that rises from left to right ,getting steeper as it persists.
Example 2
Draw the following exponential function:
y = 1/2^x
To begin, let's create a table of values.
As shown, the values of y decrease very rapidly as x increases. This is because 1/2 is less than 1.
If we were to plot the x-values and y-values on a coordinate plane, it is going to 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 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 present 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 terms are the powers of an independent variable digit. The general form of an exponential series is:
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