Investigating Exponential Functions

Collect the Data
Step 1 Cut a sheet of notebook paper in half.
Step 2 Stack the two halves, one on top of the other.
Step 3 Make a table like the one below and record
the number of sheets of paper you have in the stack after one cut.
Step 4 Cut the two stacked sheets in half, placing the resulting pieces in a single stack. Record the number of sheets of paper in the new stack after 2 cuts.
Step 5 Continue cutting the stack in half, each time putting the resulting piles in a single stack and recording the number of sheets in the stack. Stop when the resulting stack is too thick to cut.
Analyze the Data
1. Write a list of ordered pairs (x, y), where x is the number of cuts and y is the number of sheets in the stack. Notice that the list starts with the ordered pair (0, 1), which represents the single sheet of paper before any cuts were made.
2. Continue the list, beyond the point where you stopped cutting, until you reach the ordered pair for 7 cuts. Explain how you calculated the last y values for your list, after you had stopped cutting.
3. Plot the ordered pairs in your list on a coordinate grid. Be sure to choose a scale for the y-axis so that you can plot all of the points.
4. Describe the pattern of the points you have plotted. Do they lie on a straight line?
Make a Conjecture
5. Write a function that expresses y as a function of x.
6. Use a calculator to evaluate the function you wrote in Exercise 5 for x = 8 and x = 9. Does it give the correct number of sheets in the stack after 8 and 9 cuts?
7. Notebook paper usually stacks about 500 sheets to the inch. How thick would your stack of paper be if you had been able to make 9 cuts?
8. Suppose each cut takes about 5 seconds. If you had been able to keep cutting, you would have made 36 cuts in three minutes. At 500 sheets to the inch, make a conjecture as to how thick you think the stack would be after 36 cuts.
9. Use your function from Exercise 5 to calculate the thickness of your stack after 36 cuts. Write your answer in miles.

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