Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function calculates an exponential decrease or increase in a particular base. Take this, for example, let's say a country's population doubles annually. This population growth can be portrayed as an exponential function.
Exponential functions have multiple real-life use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.
Here we will learn the fundamentals of an exponential function along with important examples.
What’s the equation for an Exponential Function?
The common formula for an exponential function is f(x) = b^x, where:
-
b is the base, and x is the exponent or power.
-
b is fixed, and x varies
For instance, 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 larger than 0 and does not equal 1, x will be a real number.
How do you plot Exponential Functions?
To chart an exponential function, we must find the points where the function crosses the axes. This is known as the x and y-intercepts.
As 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, we need to set the worth for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
In following this technique, we get the domain and the range values for the function. Once we determine 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 characteristics:
-
The line crosses the point (0,1)
-
The domain is all positive real numbers
-
The range is greater than 0
-
The graph is a curved line
-
The graph is on an incline
-
The graph is smooth and continuous
-
As x approaches negative infinity, the graph is asymptomatic regarding the x-axis
-
As x advances toward positive infinity, the graph grows without bound.
In situations where the bases are fractions or decimals between 0 and 1, an exponential function exhibits the following properties:
-
The graph passes the point (0,1)
-
The range is more than 0
-
The domain is entirely real numbers
-
The graph is declining
-
The graph is a curved line
-
As x nears positive infinity, the line within graph is asymptotic to the x-axis.
-
As x approaches negative infinity, the line approaches without bound
-
The graph is smooth
-
The graph is unending
Rules
There are some essential rules to remember when working with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For instance, if we need to multiply two exponential functions that have 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 identical base, deduct the exponents.
For example, if we need to divide two exponential functions with a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To increase an exponential function to a power, multiply the exponents.
For example, if we have to raise 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 that has a base of 1 is consistently 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 regardless of what the value of x is.
Examples
Exponential functions are generally used to indicate exponential growth. As the variable increases, the value of the function grows faster and faster.
Example 1
Let’s observe the example of the growth of bacteria. Let’s say we have a culture of bacteria that multiples by two hourly, then at the end of the first hour, we will have twice as many bacteria.
At the end of the second hour, we will have quadruple as many bacteria (2 x 2).
At the end of the third hour, we will have 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be portrayed 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
Moreover, exponential functions can portray exponential decay. If we have a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of hour one, we will have half as much substance.
At the end of the second hour, we will have one-fourth as much substance (1/2 x 1/2).
At the end of hour three, we will have an 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 quantity of substance at time t and t is calculated in hours.
As shown, both of these samples use a comparable pattern, which is why they are able to be shown 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 while the base continues to be constant. This indicates that any exponential growth or decomposition where the base changes is not an exponential function.
For example, in the matter of compound interest, the interest rate stays the same while the base is static in normal time periods.
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 measure the equivalent 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 worth of y increase very quickly as x increases. If 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 continues.
Example 2
Plot the following exponential function:
y = 1/2^x
To begin, let's create a table of values.
As you can see, the values of y decrease very rapidly as x rises. This is because 1/2 is less than 1.
Let’s say we were to plot 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 descends 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 displayed as f(ax)/dx = ax. All derivatives of exponential functions display special features 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 digit. The general form of an exponential series is:
Grade Potential is Able to Help You Master Exponential Functions
If you're battling to comprehend exponential functions, or merely require a bit of extra help with math as a whole, consider working with a tutor. At Grade Potential, our Miami math tutors are experts in their subjects and can offer you with the face to face support you need to thrive.
Call us at (305) 846-7424 or contact us today to learn more about the ways in which we can assist you in reaching your academic potential.
