Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 7, Part A:
Exploring Exponential Functions

In This Part: Changing + to * | Introduction to Exponential Functions

It's amazing how different the three tables and graphs presented in Problems A1-A7 are, and all you did was change a "+" to a "*", then a "*" to a "/"!

The changes we made to the spreadsheet may have seemed small, but each change made an enormous impact in the tables and graphs. When we changed the "+" to a "*", we changed each output from a constant difference of 10 to a constant ratio of 10. This created a new type of rule called an exponential function, any function where each output is a constant multiple of the previous output.

Exponential functions often come up in real-world situations. The interest earned on an investment and the decay of nuclear waste are two good examples.

Before we move on, let's take a few moments to think about exponential notation. Just as multiplication shows repeated addition, exponents show repeated multiplication. Here are a few examples so that you can see the parallels.

You may have noticed that one example in the table above shows repeated multiplication of a fraction. Since division by a constant whole number is equivalent to multiplying by a fraction, dividing by a constant multiple also creates exponential functions.

A few terms are handy to know when you're talking about exponential functions. In the equation y = bx, b is called the base and x is called the exponent.

So far, the exponential functions we've created have used recursive rules: Each output is a multiple of the last output. As is true with linear functions, it's often more useful to write an exponential function using a closed-form rule. To do this, we'll need to use exponents.

The spreadsheet uses a ^ symbol to make exponents. Use the table below as a guide to set up the first spreadsheet of Problem A8. After typing in the rule, you should use the "Fill Down" command to copy it to the rest of the Output column.

Problem A8

Use a spreadsheet to investigate the graphs of exponential functions. For each output rule below, create an input/output table for the rule and then graph the function. Describe the graphs and how they vary for different bases, making sure to include parentheses around a fractional base when you enter these rules into a spreadsheet. Note 4

 • y = 2x • y = (2/3)x • y = (3/2)x • y = (7/10)x • y = 8x • Make up one of your own to try.

 Problem A9 Which of the graphs in Problem A8 were increasing? Which were decreasing? Explain how you could decide whether or not y = (4/5)x is increasing or decreasing without graphing it.

 The recursive rule for y is very helpful here. What changes when x grows by 1?   Close Tip The recursive rule for y is very helpful here. What changes when x grows by 1?

 Video Segment In this video segment, the class discusses the graphs of y = 2x and y = (2/3)x from Problem A8. They find rules to determine whether an exponential function will be increasing or decreasing. Watch this segment to review your work in Part A and to get a clearer understanding of why exponential functions are increasing or decreasing. Can you explain why the graph of y = (2/3)x is decreasing? You can find this segment on the session video, approximately 4 minutes and 45 seconds after the Annenberg Media logo.

 Problem A10 Is there any exponential function that increases for a while, then decreases? Is there an exponential function that never increases or decreases?

 Session 7: Index | Notes | Solutions | Video