Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or rise in a particular base. For example, let us assume a country's population doubles every year. This population growth can be portrayed in the form of an exponential function.
Exponential functions have many real-life use cases. Expressed mathematically, an exponential function is shown as f(x) = b^x.
In this piece, we will review the basics of an exponential function coupled with important examples.
What’s the formula for an Exponential Function?
The general 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 a situation where b is greater than 0 and does not equal 1, x will be a real number.
How do you graph Exponential Functions?
To plot an exponential function, we need to locate the points where the function crosses the axes. This is referred to as the x and y-intercepts.
As the exponential function has a constant, one must set the value for it. Let's take the value of b = 2.
To locate the y-coordinates, its essential to set the value for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
According to this method, we achieve the domain and the range values for the function. Once we have 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 identical qualities. When the base of an exponential function is larger than 1, the graph would have the below properties:
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The line crosses 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 smooth and constant
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As x nears 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 cases where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following qualities:
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The graph passes the point (0,1)
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The range is larger 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 continuous
Rules
There are several basic rules to bear in mind when dealing with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For example, if we have 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 the same base, subtract 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 raise 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, then we can note 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 regardless of what the value of x is.
Rule 5: An exponential function with a base of 0 is always equal to 0.
For example, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are commonly used to indicate exponential growth. As the variable increases, the value of the function increases faster and faster.
Example 1
Let’s observe the example of the growth of bacteria. If we have a group of bacteria that multiples by two every hour, then at the end of the first hour, we will have twice 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 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be displayed 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 in hours.
Example 2
Similarly, exponential functions can represent exponential decay. Let’s say we had a dangerous material that decomposes at a rate of half its volume every hour, then at the end of one hour, we will have half as much substance.
At the end of two hours, we will have a quarter as much material (1/2 x 1/2).
At the end of hour three, we will have one-eighth as much substance (1/2 x 1/2 x 1/2).
This can be represented using an exponential equation as follows:
f(t) = 1/2^t
where f(t) is the volume of substance at time t and t is measured in hours.
As shown, both of these illustrations use a similar pattern, which is why they can be shown 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 remains constant. This means that any exponential growth or decay where the base is different is not an exponential function.
For instance, in the case of compound interest, the interest rate stays the same whereas the base is static in normal amounts of time.
Solution
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we must plug in different values for x and calculate the corresponding values for y.
Let us review the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As you can see, the values of y increase very rapidly as x grows. Imagine we were to draw 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 and gets steeper as it persists.
Example 2
Graph 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 swiftly as x rises. The reason 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:
This is a decay function. As shown, the graph is a curved line that decreases from right to left and gets smoother as it continues.
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 exhibit unique characteristics by which 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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