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 annually. This population growth can be depicted in the form of an exponential function.
Exponential functions have numerous real-life uses. In mathematical terms, an exponential function is shown as f(x) = b^x.
Today we will review the essentials of an exponential function in conjunction with relevant examples.
What’s the formula for an Exponential Function?
The generic 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 fixed, and x varies
As an illustration, if b = 2, then we 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 the event where b is higher than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To chart an exponential function, we must find the dots where the function crosses the axes. This is referred to as 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 locate the y-coordinates, its essential to set the rate for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
According to this approach, we get the domain and the range values for the function. Once we have 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 similar characteristics. When the base of an exponential function is more than 1, the graph is going to have the following characteristics:
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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 greater 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 approaches negative infinity, the graph is asymptomatic concerning the x-axis
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As x nears positive infinity, the graph increases without bound.
In situations where the bases are fractions or decimals in the middle of 0 and 1, an exponential function displays the following characteristics:
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The graph crosses 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 declining
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The graph is a curved line
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As x advances toward positive infinity, the line within 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 smooth
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The graph is unending
Rules
There are several basic rules to recall when dealing 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 with a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, deduct the exponents.
For example, if we need to divide two exponential functions with 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 example, 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 with a base of 1 is consistently equal to 1.
For example, 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 no matter what the value of x is.
Examples
Exponential functions are generally used to signify exponential growth. As the variable grows, the value of the function grows quicker and quicker.
Example 1
Let’s examine the example of the growing of bacteria. Let us suppose that we have a culture of bacteria that multiples by two every hour, then at the close of the first hour, we will have double as many bacteria.
At the end of hour two, we will have 4 times 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 an exponential function as follows:
f(t) = 2^t
where f(t) is the amount of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can portray exponential decay. Let’s say we had a dangerous material that degenerates at a rate of half its amount every hour, then at the end of hour one, we will have half as much substance.
After the second hour, we will have a quarter as much material (1/2 x 1/2).
After three hours, we will have one-eighth 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 quantity of material at time t and t is calculated in hours.
As you can see, both of these illustrations follow a similar pattern, which is why they can be depicted 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 denoted by the variable whereas the base stays the same. 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 while the base changes in regular time periods.
Solution
An exponential function is able to be graphed utilizing a table of values. To get the graph of an exponential function, we need to enter different values for x and then calculate the corresponding 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 values of y grow very quickly as x rises. 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 goes up from left to right and gets steeper as it persists.
Example 2
Draw the following exponential function:
y = 1/2^x
To begin, let's make a table of values.
As you can see, the values of y decrease very swiftly as x increases. 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 the following:
This is a decay function. As shown, the graph is a curved line that gets lower 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 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 terminology are the powers of an independent variable digit. The common form of an exponential series is:
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