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Exponential Functions

 Activity Description Activity Guide

(1)  Place two sheets of legal-sized paper (Sheet 1 and Sheet 2) on your desk. You are going to fold the two sheets of paper in various ways and record the number of rectangles present on each sheet after each fold (See note below). Start with Sheet 1. Record the number of rectangles, in Table 1, before any folds are made in the paper. Now fold the paper in half. Unfold the paper, count the number of rectangles, and record the result.

 Note: Only count individual rectangles made by the current fold.  Do not count rectangles made by combining more than one rectangle. Example:  After one fold the paper should look like the diagram below. There are three rectangles shown, namely ACDF, ABEF, and BCDE.   However, only rectangles ABEF and BCDE should be counted after the paper is folded one time.

Continue folding into successive halves, keeping track of the number of rectangles constructed after each fold. Complete Table 1 after you make several folds. Describe any patterns you see in the table. Predict the number of rectangles for seven folds; for n-folds.

 Number of Folds Number of Rectangles 0 1 2 3 4

Table 1

(2)  Repeat task 1 with Sheet 2, but instead of folding the paper in halves, tri-fold the paper like a pamphlet (Tri-fold 1). Unfold the paper and count the rectangles, as shown in the figure below, and then refold the paper.

 Note:  After one fold the paper should look like the figure below.

Construct Tri-fold 2 by tri-folding Tri-fold 1. Open up Tri-fold 2 and count the number of rectangles.  Refold the paper. Continue folding into successive thirds, keeping track of the number of rectangles constructed after each tri-fold.  Complete Table 2 after you make several tri-folds. Describe any patterns you see in the table.  Predict the number of rectangles for seven tri-folds; for n-tri-folds.

 Number of Tri-folds Number of rectangles 0 1 2 3 4

Table 2

(3)  Using the data in Table 1 and Table 2, construct scatterplots for each set of data on the same graph.  Describe the characteristics of each scatterplot. What similarities do the scatterplots possess? What are some differences between them? Predict the coordinates of the next data point for each scatterplot.

 Note:  It may be convenient to use the statistical features of a graphing calculator to construct the scatterplots.

(4)  Determine if any of the following classes of functions can be manipulated to perfectly model the data in each scatterplot:  linear functions, quadratic functions, exponential functions. Discuss your results.  Compare your 'best fit' model for each scatterplot with others in the class.

 Note:  You can use the regression feature of a graphing calculator as an aid for completing  task 4.

When the activity loads it will look like the screenshot below.

 y = Makx

(6)  Notice the form of exponential functions (y = Makx), where ‘a’ denotes the base, ‘M’ denotes the leading coefficient, and ‘k’ denotes the exponential coefficientUsing the sliders, manipulate ‘M’, ‘k’, and ‘a’ so that the equation and graph model the data from Table 1  (Note:  To manipulate the slider, you can either click and drag on the slider or click on the value to the right of the slider, type in the desired value and then press enter.). Relate the value of each variable in the resulting equation to the data in Table 1.  How do these values change for the data in Table 2?

(7)  Predict the shape of the graph when ‘k’ is equal to zero. Set ‘k’ equal to zero by typing in 0 (zero) and pressing the enter key. Observe the graph and assess your prediction.  Manipulate the ‘a’ slider.  What happens to the graph? Now manipulate the ‘M’ slider. What happens to the y-intercept of the graph as ‘M’ changes values? Make conjectures as to why this happensWhat can you say about a base raised to the zero power?

(8)  By manipulating the sliders, determine the conditions necessary to have the graph traverse into the third and fourth quadrant.

(9)  Set M =1, a = 2, and k = 1. What happens to the graph as x approaches negative infinity? Does the graph have an x-intercept?  What is the range of y-values when x is negative?  Explain why the y-values are in this range. Manipulate the ‘M’ and ‘a’ sliders to determine when the range of y-values change for negative x-values.

(10)  Set M =1, a =3, and k = 2. Change the value of ‘M’ from 1 to –1. Compare the graphs of y = a2x to  y = -a2x.  What can be said about the relationship between these two graphs? Under what conditions is  y = Bakx  the reflection of  y = Cakx about the x-axis? Test your conjectures by manipulating the sliders.

(11)  Set M =2, a =3, and k = 1. Change the value of ‘k’ from 1 to –1. Compare the graphs of
y = (2)3(x) to y = (2)3(-x).  What can be said about the relationship between these two graphs?  Under what conditions is y = Mabx  the reflection of y = Macx about the y-axis? Test your conjectures by manipulating the sliders.

(12)  Redo Part 1 of the activity.  This time, however, count all the rectangles present after each fold regardless of dimension.  Determine equations that model the resulting sequences. Note: there are six rectangles in the figure below.

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