Exponential funtions

GO Exponential Functions Exponential functions, while similar to functions involving exponentsare different because the variable is now the power rather than the base. Before, we dealt with functions of the form where the variable x was the base and the number was the power.

Exponential Functions An exponential function is of the form: In this section, we will look at the domain and range of these exponential functions, as well as, look at one specific exponential function, compound interest. The domain of a function are the possible x-values while the range are the possible y-values.

We will find the domain and range by looking at the graphs of some exponential functions. In order to graph the function, f x was replaced with y. Now, based on the graph we want to find the domain and range.

Since x is an exponent and the exponent can be any real number the domain is all real numbers which is written as: So, this makes the graph look like it stops at -5 when in reality it has just gotten so close to the x-axis that the computer is unable to differentiate between the two.

Exponential function | Easy Math

Now, we need to find the range. The range are the y-values. Since the graph never intersects or goes below the x-axis, the y-values are not zero or negative.

This makes the range, all positive real numbers. This is written as: Here, having a decimal less than 1 verses a whole number greater than one does not change the domain and range.

This downward slant is known as exponential decay. What happens if the exponent is negative? Again, the domain and range did not change. All this did was change the direction of the graph. The original graph which is in Example I went upward from left to right where as adding the negative to the exponent made the graph go downward from left to right.

What about if the exponential function is multiplied by a cosntant?

Discussion

It appears to be very similar to the original. Do you see the difference? Have you figured out what that constant does? If not, play the following movie and see if it helps you to figure out what the constant does to the graph.

The original graph will be in purple and the graph in red shows what happens as the value in front of the exponential function changes. Have you figured it out? If so, write a brief one page explanation worth 10 bonus points. The last idea that we will look at is that of compound interest.

That is the total amount of an investment, A, earning compound interest is where P is the pricipal, r is the annual interest rate, n is t he number of times interest is compounded per year, and t is the time in years.

These problems will be worked as a straight substitution and are thus very simple to solve.The inverse of the exponential function is the logarithmic function or logarithm. This general logarithm is written as y = a log(x) or y = log a (x).

For e as the base of the logarithm the curve is called the natural logarithm ln(x) 2). Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. Exponential functions are functions of the form f(x) = b^x where b is a constant. The real mathematical importance of exponential functions is in their being proportional to their derivatives meaning the bigger x is, the steeper the slope of the function.

the important elementary function f(z) = e z; sometimes written exp z. It is encountered in numerous applications of mathematics to the natural sciences and engineering.

For any real or complex value of z, the exponential function is defined by the equation. It is obvious that e 0 = 1.

When z = 1, the value of the function is equal to e, which is the base of the system of natural logarithms. Thus, exponential functions have a constant base; the variable is in the exponent. The number $\,b\,$ is called the base of the exponential function. The most important exponential function is when the base is the irrational number $\,\text{e}\,$.

exponential functions by making a table of values and plotting the points. After graphing the first two examples we will take a look at the s imilarities and differences between the two graphs.

Exponential Functions Gizmo : Lesson Info : ExploreLearning