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Exponential Functions


 
Part 1: Paper folding activity (1) Place two sheets of legalsized 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.
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 nfolds.
Table 1 (2) Repeat task 1 with Sheet 2, but instead of folding the paper in halves, trifold the paper like a pamphlet (Trifold 1). Unfold the paper and count the rectangles, as shown in the figure below, and then refold the paper.
Construct Trifold 2 by trifolding Trifold 1. Open up Trifold 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 trifold. Complete Table 2 after you make several trifolds. Describe any patterns you see in the table. Predict the number of rectangles for seven trifolds; for ntrifolds.
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.
(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.
Part 2: Features of exponential graphs (5) Point your web browser to ExploreMath’s “Exponential Functions” Activity located at: http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=4 When the activity loads it will look like the screenshot below.
(6) Notice the form of exponential functions (y = Ma^{kx}), where ‘a’ denotes the base, ‘M’ denotes the leading coefficient, and ‘k’ denotes the exponential coefficient. Using 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 yintercept of the graph as ‘M’ changes values? Make conjectures as to why this happens. What 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 xintercept? What is the range of yvalues when x is negative? Explain why the yvalues are in this range. Manipulate the ‘M’ and ‘a’ sliders to determine when the range of yvalues change for negative xvalues. Part 3: Reflections and exponential graphs (10) Set M =1, a =3, and k = 2. Change the value of ‘M’ from 1 to –1. Compare the graphs of y = a^{2x} to y = a^{2x}. What can be said about the relationship between these two graphs? Under what conditions is y = Ba^{kx} the reflection of y = Ca^{kx} about the xaxis? 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 (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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