Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a certain 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 many real-life applications. In mathematical terms, an exponential function is displayed as f(x) = b^x.
Today we will review the fundamentals of an exponential function coupled with relevant examples.
What’s the formula for an Exponential Function?
The general 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 greater than 0 and does not equal 1, x will be a real number.
How do you chart Exponential Functions?
To plot an exponential function, we need to find the spots where the function intersects the axes. These are referred to 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 discover the y-coordinates, its essential to set the value for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
According to this method, we achieve the range values and the domain for the function. After having the rate, we need to graph 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 more than 1, the graph is going to have the below qualities:
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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 larger than 0
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The graph is a curved line
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The graph is rising
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The graph is level 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 approaches positive infinity, the graph rises without bound.
In situations where the bases are fractions or decimals between 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 more 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 advances toward positive infinity, the line in the graph is asymptotic to the x-axis.
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As x approaches negative infinity, the line approaches without bound
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The graph is level
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The graph is unending
Rules
There are a few essential rules to remember when engaging with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For instance, if we have to multiply two exponential functions that have 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, deduct the exponents.
For instance, if we have to divide two exponential functions that posses a base of 3, we can write it as 3^x / 3^y = 3^(x-y).
Rule 3: To grow an exponential function to a power, multiply the exponents.
For instance, if we have to increase an exponential function with a base of 4 to the third power, then we can write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equivalent to 1.
For instance, 1^x = 1 regardless of what the rate 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 usually used to denote exponential growth. As the variable rises, the value of the function grows at a ever-increasing pace.
Example 1
Let’s observe the example of the growth of bacteria. If we have a culture 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 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 represented using an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured hourly.
Example 2
Also, exponential functions can represent exponential decay. If we have a radioactive material that degenerates at a rate of half its quantity every hour, then at the end of one hour, we will have half as much substance.
After hour two, we will have one-fourth as much material (1/2 x 1/2).
After three hours, we will have 1/8 as much substance (1/2 x 1/2 x 1/2).
This can be displayed using an exponential equation as below:
f(t) = 1/2^t
where f(t) is the amount of substance at time t and t is measured in hours.
As demonstrated, both of these examples use a comparable pattern, which is why they can be represented using exponential functions.
As a matter of 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 denoted by the variable while the base remains constant. This indicates that any exponential growth or decline where the base changes is not an exponential function.
For instance, in the scenario of compound interest, the interest rate continues to be the same while the base varies in ordinary amounts of time.
Solution
An exponential function can be graphed employing 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's check out the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As demonstrated, the values of y rise very rapidly as x grows. If we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As you can see, the graph is a curved line that goes up from left to right ,getting steeper as it goes.
Example 2
Plot 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 swiftly as x rises. This is because 1/2 is less than 1.
If we were to graph the x-values and y-values on a coordinate plane, it is going to look like what you see below:
The above is a decay function. As shown, the graph is a curved line that decreases 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 shown as f(ax)/dx = ax. All derivatives of exponential functions display special features 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 expressions are the powers of an independent variable figure. The common form of an exponential series is:
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