Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates 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 represented in the form of an exponential function.
Exponential functions have many real-world uses. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Today we will review the fundamentals of an exponential function coupled with important 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 cases where b is higher than 0 and unequal to 1, x will be a real number.
How do you plot Exponential Functions?
To plot an exponential function, we must locate the points where the function intersects the axes. This is known as the x and y-intercepts.
Since the exponential function has a constant, it will be necessary to set the value for it. Let's focus on the value of b = 2.
To locate the y-coordinates, its essential to set the rate for x. For instance, for x = 1, y will be 2, for x = 2, y will be 4.
In following this method, we achieve 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 comparable properties. When the base of an exponential function is greater than 1, the graph will have the following properties:
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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 greater 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 ongoing
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As x advances toward negative infinity, the graph is asymptomatic towards the x-axis
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As x approaches positive infinity, the graph grows without bound.
In cases where the bases are fractions or decimals within 0 and 1, an exponential function displays the following qualities:
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The graph passes 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 nears 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 smooth
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The graph is constant
Rules
There are several basic rules to remember when working 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 posses a base of 2, then we can compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an equivalent base, subtract the exponents.
For example, if we need to divide two exponential functions that posses a base of 3, we can note 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 note it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function with 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 identical to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are generally used to indicate exponential growth. As the variable grows, the value of the function rises at a ever-increasing pace.
Example 1
Let’s examine the example of the growth of bacteria. Let us suppose that we have a culture of bacteria that duplicates each 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 total sum of bacteria at time t and t is measured hourly.
Example 2
Also, exponential functions can illustrate exponential decay. Let’s say we had a radioactive 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 material.
After hour two, we will have 1/4 as much material (1/2 x 1/2).
After hour three, we will have one-eighth as much material (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 amount of material at time t and t is assessed in hours.
As shown, both of these examples follow a comparable pattern, which is the reason they can be depicted using exponential functions.
In 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 continues to be fixed. This indicates that any exponential growth or decay where the base varies is not an exponential function.
For instance, in the case of compound interest, the interest rate remains the same while the base varies in normal time periods.
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 enter different values for x and asses the matching values for y.
Let us review this example.
Example 1
Graph the this exponential function formula:
y = 3^x
First, let's make a table of values.
As shown, the values of y increase very rapidly as x grows. If we were to plot this exponential function graph on a coordinate plane, it would look like this:
As seen above, the graph is a curved line that rises from left to right ,getting steeper as it continues.
Example 2
Graph the following exponential function:
y = 1/2^x
To start, let's create a table of values.
As shown, the values of y decrease very quickly 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 would 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 flatter as it goes.
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 unique properties whereby 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 terminology are the powers of an independent variable digit. The general form of an exponential series is:
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