Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a specific base. 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 applications. Expressed mathematically, an exponential function is shown as f(x) = b^x.
Today we will review the basics of an exponential function along 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 fixed, and x varies
For example, 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 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 need to discover the points where the function intersects 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, we need 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 get the range values and the domain for the function. Once we have the worth, we need to graph them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable 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 on an incline
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The graph is flat and continuous
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As x advances toward negative infinity, the graph is asymptomatic regarding the x-axis
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As x advances toward positive infinity, the graph rises without bound.
In instances where the bases are fractions or decimals within 0 and 1, an exponential function presents with the following characteristics:
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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 entirely real numbers
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The graph is decreasing
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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 gets closer to 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 vital rules to recall when working 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 compose it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an equivalent base, deduct the exponents.
For instance, if we need to divide two exponential functions that have a base of 3, we can compose it as 3^x / 3^y = 3^(x-y).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For example, 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 with a base of 1 is forever 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 equivalent to 0.
For instance, 0^x = 0 despite whatever the value of x is.
Examples
Exponential functions are generally utilized to denote exponential growth. As the variable increases, the value of the function rises at a ever-increasing pace.
Example 1
Let's look at the example of the growing of bacteria. If we have a cluster of bacteria that doubles hourly, then at the end of hour one, we will have double 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 represented using 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
Moreover, exponential functions can portray exponential decay. Let’s say we had a dangerous material that decomposes at a rate of half its amount every hour, then at the end of hour one, we will have half as much material.
At the end of the second hour, we will have one-fourth 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 below:
f(t) = 1/2^t
where f(t) is the quantity of substance at time t and t is measured in hours.
As you can see, both of these examples use a comparable pattern, which is the reason they can be shown using exponential functions.
As a matter of fact, any rate of change can be indicated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base stays fixed. This means that any exponential growth or decline where the base changes is not an exponential function.
For example, in the scenario of compound interest, the interest rate continues to be the same whilst the base is static 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 must enter different values for x and calculate the matching values for y.
Let's look at the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As demonstrated, the rates of y rise very fast as x rises. 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 rises from left to right ,getting steeper as it persists.
Example 2
Graph the following exponential function:
y = 1/2^x
To begin, let's draw up a table of values.
As shown, the values of y decrease very swiftly as x surges. This is because 1/2 is less than 1.
Let’s say we were to graph 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 descends 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 written as f(ax)/dx = ax. All derivatives of exponential functions display particular properties whereby 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 number. The general form of an exponential series is:
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