Most functions we have looked at so far have x as the base and some number as the exponent of x. For instance, f(x)=x2, is very common. Now let’s switch it up.
Exponential functions take on the form of f(x)=abx and represent growth or decay in the real world. Because the variable of x is the exponent, as x gets larger and larger (or smaller and smaller), the function grows (or shrinks) “exponentially.”
Let’s try to break this down so we can understand what’s happening.
Components of an exponential function.
The norm in algebra is to raise a variable to a specific power. The example used above is f(x)=x2.
Before we talk about exponential function, let’s look at its parts.
The exponential function can consist of three parts. These parts are the coefficient, base, and exponent.
The only two that are necessary are the base and the exponent. Let’s compare the base of a polynomial to the base of an exponential function.
In a polynomial, our base would be the variable.
In an exponential function, the base is a constant. This number must be a positive number.
The exponent in a polynomial can be any real number. It remains constant in a polynomial. But in an exponential function, the exponent is what changes. It is the independent variable in our function.
Properties and rules of exponential functions.
If you know the properties or rules of exponents, each rule can easily translate to exponential functions.
We know that if we want to combine two expressions that contain exponents, both expressions must contain the same base. We can’t combine the exponents of x2 with y3. If we multiply the two expressions, x2*y3, we only get x2y3.
To combine two different expressions, the bases must be identical. If we have x2*x3, we can easily combine the two since the bases are identical. In this case, the answer is x5. This same principle applies to exponential functions.
I created a chart of properties that are similar between exponents and exponential functions. The first chart shows the rules or properties of exponents. The chart after that shows how those rules relate to exponential functions.
What does the graph of an exponential function look like?
Before we look at the graph, let’s look at what happens to an exponential function when different parts of the function change.
The base and the exponents reveal a lot about how the exponential function behaves. It shows the rate of change and the direction of change for each function. Both components show how quickly or slowly the function increases or decreases.
What happens to an exponential function’s graph if the base is a value between 0 and 1? As x increases, the function’s value falls. Here’s a table representing the values for the exponential function f(x)=.5x. As the x values increase, the function itself approaches zero. There is a decrease in the function’s values. This is called exponential decay.
What happens to the exponential function’s graph when the base is greater than 1? As x increases, the function’s values rise. As the values get larger and larger, the function’s values get larger and larger. Here is another table showing what happens to the exponential function f(x)=5x. This is referred to as exponential growth.
Now let’s take a look at what happens when different parts of the function are changed in different ways.
What happens to our function when the base is greater than 1 but the exponents decrease instead? The function’s value gets closer and closer to zero as the x value decreases. We can see this again with the function f(x)=5x.
The values for this graph can be seen in the graph above for f(x)=5x. Our focus this time is on the values as they move from right to left instead of left to right. As the values go from right to left, the function’s values decrease.
Let’s try to adjust the graph in another way. If we have a coefficient in front of the exponential expression, what happens to the values of the function?
The graph compresses when the coefficient in an exponential function is between 0 and 1.
The graph stretches out when the coefficient in an exponential function is greater than 1.
When the coefficient is negative, the resulting exponential function is reflected across the y-axis.
When we add or subtract a value to the exponential expression, the function shifts vertically up or down.
When we add or subtract a value to the exponent, the function shifts horizontally, left or right.
If we multiply our exponent by a number greater than one, we see that the graph rises quicker than the original. The exponential function f(x)=2(2x) grows twice as fast as the original.
If we multiply our exponent by a number between zero and one, we find that our function increases at a slower rate than the original. The exponential function f(x)=2(.5x) rises slower than the original function.
Most common exponential functions: e and 10.
Although any positive number can be used as a base in exponential functions, the two most commonly used are e and 10.
The base 10 number system is the most familiar counting system. Number bases are the number of digits that a counting system uses to show numbers. As human beings, we use our ten fingers to count.
The base 10 system is also referred to as the decimal system. In the decimal system, a digit’s value is determined by where it is in relation to the decimal point. For instance, the number 6,345 has a 3 in the hundreds position. In the number 3.546, the 4 is in the hundredth position.
The base e is a bit harder to explain. The “e” of base e is known as Euler’s number. This number is a mathematical constant whose value is about 2.71828. In real life, this value is a nonrepeating number that goes on forever, like Pi. It is useful in describing continuous growth or decay. We’ll look at more example problems later on.
Later, we’ll look at not only how to solve exponential equations but also how to graph them.
Keep up the good work!