Exponential Functions  Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function calculates an exponential decrease or rise in a particular base. For instance, let us assume a country's population doubles every year. This population growth can be depicted as an exponential function.
Exponential functions have many reallife uses. Expressed mathematically, an exponential function is shown as f(x) = b^x.
In this piece, we discuss the fundamentals of an exponential function along with important examples.
What is 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 is a variable
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 cases where b is greater than 0 and not equal to 1, x will be a real number.
How do you chart Exponential Functions?
To graph an exponential function, we have to locate the spots where the function intersects the axes. This is called the x and yintercepts.
Considering the fact that 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 ycoordinates, one must to set the worth for x. For example, for x = 2, y will be 4, for x = 1, y will be 2
By following this approach, we determine the range values and the domain for the function. Once we determine the rate, we need to graph them on the xaxis and the yaxis.
What are the properties of Exponential Functions?
All exponential functions share identical characteristics. When the base of an exponential function is greater than 1, the graph would have the following qualities:

The line crosses the point (0,1)

The domain is all positive real numbers

The range is more than 0

The graph is a curved line

The graph is rising

The graph is level and constant

As x advances toward negative infinity, the graph is asymptomatic regarding the xaxis

As x approaches positive infinity, the graph grows without bound.
In situations where the bases are fractions or decimals between 0 and 1, an exponential function displays the following properties:

The graph intersects the point (0,1)

The range is more than 0

The domain is entirely real numbers

The graph is descending

The graph is a curved line

As x advances toward positive infinity, the line in the graph is asymptotic to the xaxis.

As x gets closer to negative infinity, the line approaches without bound

The graph is flat

The graph is constant
Rules
There are some essential rules to recall when engaging with exponential functions.
Rule 1: Multiply exponential functions with the same base, add the exponents.
For instance, if we need to multiply two exponential functions that have a base of 2, then we can write it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with the same base, subtract the exponents.
For example, if we have to divide two exponential functions that posses a base of 3, we can compose it as 3^x / 3^y = 3^(xy).
Rule 3: To raise an exponential function to a power, multiply the exponents.
For instance, if we have to grow an exponential function with a base of 4 to the third power, we are able to compose 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 value of x is.
Rule 5: An exponential function with a base of 0 is always identical to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Examples
Exponential functions are usually leveraged to denote exponential growth. As the variable grows, the value of the function rises quicker and quicker.
Example 1
Let’s examine the example of the growing of bacteria. Let’s say we have a group of bacteria that multiples by two each hour, then at the close of hour one, 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 hour three, we will have 8x 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 amount of bacteria at time t and t is measured hourly.
Example 2
Moreover, exponential functions can portray exponential decay. If we have a radioactive material that decomposes at a rate of half its quantity every hour, then at the end of hour one, we will have half as much substance.
After two hours, we will have 1/4 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 volume of substance at time t and t is measured in hours.
As shown, both of these samples pursue a comparable pattern, which is why they are able to be represented using exponential functions.
In fact, any rate of change can be denoted using exponential functions. Bear in mind that in exponential functions, the positive or the negative exponent is depicted by the variable while the base continues to be the same. This means that any exponential growth or decay where the base is different is not an exponential function.
For instance, in the case of compound interest, the interest rate remains the same while the base is static in regular amounts of time.
Solution
An exponential function is able to be graphed using a table of values. To get the graph of an exponential function, we must input different values for x and then calculate the equivalent values for y.
Let's look at the following example.
Example 1
Graph the this exponential function formula:
y = 3^x
To begin, let's make a table of values.
As shown, the worth of y rise very quickly as x grows. 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 ,getting steeper as it continues.
Example 2
Draw the following exponential function:
y = 1/2^x
To begin, let's make a table of values.
As shown, the values of y decrease very quickly as x increases. The reason is because 1/2 is less than 1.
Let’s say we were to plot the xvalues and yvalues on a coordinate plane, it would look like what you see below:
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 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 exhibit special characteristics 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 terminology are the powers of an independent variable figure. The general form of an exponential series is:
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