Exponential Functions - Formula, Properties, Graph, Rules
What’s an Exponential Function?
An exponential function measures an exponential decrease or rise in a specific base. For example, let us suppose a country's population doubles every year. This population growth can be portrayed as an exponential function.
Exponential functions have many real-life uses. Mathematically speaking, an exponential function is shown as f(x) = b^x.
In this piece, we will learn the fundamentals of an exponential function along with important examples.
What’s 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 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 a situation where b is greater than 0 and does not equal 1, x will be a real number.
How do you graph Exponential Functions?
To plot an exponential function, we must locate the dots 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 focus on the value of b = 2.
To locate the y-coordinates, one must to set the rate for x. For instance, for x = 2, y will be 4, for x = 1, y will be 2
By following this method, we determine the domain and the range values for the function. After having the worth, we need to draw them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share similar properties. When the base of an exponential function is larger than 1, the graph will have the following 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 rising
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The graph is level and continuous
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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 rises without bound.
In events where the bases are fractions or decimals between 0 and 1, an exponential function displays the following properties:
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The graph passes the point (0,1)
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The range is greater than 0
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The domain is entirely 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 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 flat
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The graph is constant
Rules
There are several 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 posses a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an equivalent base, deduct 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^(x-y).
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 that has a base of 1 is consistently equal 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 equivalent to 0.
For example, 0^x = 0 no matter what the value of x is.
Examples
Exponential functions are commonly leveraged to indicate exponential growth. As the variable grows, the value of the function grows at a ever-increasing pace.
Example 1
Let’s observe the example of the growth of bacteria. Let us suppose that we have a group of bacteria that multiples by two every hour, then at the close of hour one, we will have double as many bacteria.
At the end of the second hour, we will have 4 times 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 portrayed 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 in hours.
Example 2
Similarly, exponential functions can illustrate exponential decay. If we have a radioactive substance that degenerates at a rate of half its volume every hour, then at the end of hour one, we will have half as much material.
After the second hour, we will have 1/4 as much substance (1/2 x 1/2).
At the end of hour three, we will have one-eighth as much material (1/2 x 1/2 x 1/2).
This can be shown 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 measured in hours.
As you can see, both of these samples use a similar 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 represented by the variable whereas the base continues to be the same. Therefore any exponential growth or decomposition where the base changes is not an exponential function.
For example, in the scenario of compound interest, the interest rate stays the same whilst the base changes in normal 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 plug in different values for x and then asses the corresponding values for y.
Let us check out the example below.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As shown, the rates of y grow very rapidly as x rises. 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 goes up from left to right and gets steeper as it continues.
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 quickly as x surges. 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 this:
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 particular properties by which 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 figure. The common form of an exponential series is:
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