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Exploring Period...Period.

 Activity Description Activity Guide

## Part 1:Exploring Periodic Graphs

1.)  Predict what the following graph would look like if the pattern was repeated indefinitely. Draw your prediction on the graph. (see Handout 1).

2.)  Divide your graph using dashed lines into equal intervals so that the partitions are identical; that is, the portion of graph inside each partition looks exactly the same as the partitions adjacent to it. (An example is shown below.)  What is the width of each of your partitions?  For example, in the diagram below, the width of each partition is 2 units, or four tick marks.

3.)  For the graph in #1, what is the minimum partition width you could use and still ensure that the partitions are identical?

4.)  Put a piece of tracing paper over the graphs below and partition the graph into identical partitions using the minimum partition width. Compare your answer with two neighbors.  Predict how many distinct ways you could construct identical partitions using the minimum partition width.  Show at least two different ways to partition the graph into identical partitions using the minimum partition width. (see Handout 2)

5.)  Put your piece of tracing paper with its partition lines over the graph so that you have identical partitions as in Task 4.  Now, shift the tracing paper to the left one-half unit.  (Do not redraw or shift the graph.  Only shift the tracing paper on top of the graph.)  Compare the portions of graph inside each shifted partition to the partition adjacent to it.

6.)  Shift the tracing paper to the right one and one-fourth unit.  Compare the portions of graph inside each shifted partition to the partition adjacent to it and discuss your findings.

7.)  Now how many ways do you think you could construct identical partitions using the minimum partition width?  Describe the ways you have discovered.

8.)  The minimum partition width used to construct identical partitions is referred to as the length of one period or one cycle. In fact, we say that graphs like this one are periodic or cyclic.  Why?

Part 2: Exploring the Graph of y = sin(x) and y = cos(x).

Using your web browser go to the “Shifting and scaling sine and cosine curves” Activity located at http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=23. When the activity loads it will look like the following.

1.)  Notice that the graph of y = sin(x) is shown. By looking at its graph, determine if the sine function is periodic. If it is, determine the length of one period.

 To help you with your decision, you might to “zoom out” and “zoom in” by clicking on the magnifying glasses containing the minus and plus signs. You can also pan left or pan right by clicking on the red arrows pointing to the left and the right.

2.)  Manipulate the red arrows and the magnifying glasses until the first x-intercept you see on the graph is at 0o and the last x-intercept you see is at 360o.

3.)  Over the interval [0°, 360°], what are the minimum and maximum y-values of y = sin(x)? Find the global minimum and maximum y-values over the entire domain of real numbers. Explain and compare your answers with your neighbors.

4.)  How far apart are the x-intercepts of y = sin(x) on the interval [0°, 360°]? Generalize your findings for the entire domain of y = sin(x).

 Check your answers and predictions by using the red arrow buttons or zooming out and viewing as much of the full graph of y = sin(x) as you wish. Alternatively, you may verify your answers numerically by utilizing the “Calculate data values” clipboard tool to see a table of values.

5.)  Repeat the tasks in Part 2 for y = cos(x).

# Part 3: The Variable b in the Equation y = a sin[b(x-c)] + d

The general form of the sine equation is:

y = a sin[b(x-c)] + d where a, b, c, d € R.

By substituting different numbers for a, b, c, and d, we can shift and scale the graph of the sine function.

1.)  Write the sine equation for a = 1, c = 0o, and d = 0.  What does this graph look like when b = 1?

 Teacher Note: The next activities have students explore the graph of the sine function when  b =/ 1. Activities #2 and #3 show how the value chosen for the variable b affects the graph at x = 0.

2.)  Algebraically determine the values of the variable b for which (0,0) is a solution to the equation
y = sin(bx).

3.)  In the ExploreMath activity, manipulate the b-slider and watch what happens to the graph of
y = sin(bx) for different values of b at x = 0. How do the graphical representations relate to the algebraic solutions?

 In the ExploreMath activity, to center the graph about the origin, click on the red circle that is located between the red arrows.

4.)  Click on the red circle if necessary and the magnifying glasses to make -360o the first labeled tic mark on the x-axis and 360o the last. Set b to 0.2 by clicking on the number next to the b-slider, typing in the number 0.2, and hitting enter. Now, slowly move the b-slider to the right. What happens to the graph as b gets larger?

5.)  From what you have seen, make a conjecture about the relationship between b and the length of one period.

6.)  To test your conjecture, repeat the instructions in #4, but this time click on the box marked “Show amplitude, period, frequency” in the ExploreMath activity. As you move the b-slider slowly to the right from 0.2, watch the value of the period change. Was your conjecture correct?

7.)  From what you have seen, make conjectures about:
a.)  the length of one period as b approaches 0, and
b.)  the appearance of the full graph when b equals 0.

8.)  Test your conjectures in #7 by setting b equal to 0.2, and then watching the changes in the graph and the changes in the period as you enter decreasing values of b, e.g., 0.1, 0.05, 0.03, 0.01, and 0.

 To get a good look at what is happening in #8, center the graph using the red circle and zoom out as far as possible using the ‘-’ magnifying glass. Then, click on the right red arrow until 2880o is the last labeled tic mark you see on the x-axis.  The graph should look like the one below.

9.)  Algebraically test your conjecture in #7b by noticing the type of equation that results when you substitute 0 for b into the equation y = sin(bx).

# Part 4: Taking a Closer Look at the Variable b

1.)  Fill in the table below for y = sin(bx) when b = 1. Then, use the table and the graph below to sketch one cycle of sine starting at the origin. (see Handout 3.)

 x y=sin(x) 0o 90o 180o 270o 360o

2.)  Fill in the tables for y = sin(bx) when b = 2 and for y = sin(bx) when b = 4. As you do, predict how the graphs of y = sin(2x) and y = sin(4x) will be different from the graph of y = sin(x).

3.)  Check your conjectures by drawing the graphs of one cycle of y = sin(2x) and one cycle of
y =  sin(4x) on the same axes as y = sin(x). What is the relationship between one period of
y = sin(2x), one period of y = sin(4x), and one period of y = sin(x)?

 x 2x y = sin(2x) 0o 90o 180o 270o 360o
 x 4x y = sin(4x) 0o 90o 180o 270o 360o

4.)  Predict the length of one cycle of y = sin(0.5x). How does this length compare to the length of one cycle of y = sin(x)? Test your prediction using the ExploreMath activity by sliding b to 0.5 and looking at the graph and its period.

5.)  Derive a formula for the period of y = sin(bx) when b > 0.  Using a variety of b values, test the validity of your formula algebraically and using ExploreMath.

# Part 5:  The Effect of Negative Values of b

1.)  Zoom out so that -720 is the last labeled tic mark on the x-axis. Set b to 0 by clicking on the number next to the b-slider, typing in the number 0, and hitting enter. Now, slowly move the b-slider to the left. What happens to the period as b gets larger in the negative direction? How does this compare/contrast to what happens to the period as b gets larger in the positive direction?

 Note: Be careful not to confuse period with frequency. Frequency refers to the number of cycles that occur in a certain interval. What happens to the frequency as b gets larger in the positive direction? In the negative direction?

2.)  What kind of relationship exists between the period and the magnitude of b; that is, direct, inverse, or no relationship? Hint: magnitude of b = |b|).

 Remember that as b gets larger in the positive direction, the period and b have an inverse relationship. Since the magnitude of any b-value always equals a b-value that is positive (or equal to zero), we know that the period and the magnitude of b have an inverse relationship as well.

Next, we’d like to know if the graph of the sine function for a particular positive value of b corresponds to the graph of the sine function for the opposite of that particular value of b. For example, we’d like to know how the graph of y = sin(2x) compares to the graph of y = sin(-2x).

3.)  Complete the tables below for y = sin(-2x) and y = sin(-4x) without using a calculator or graph. (Hint: Use the fact that the sine function is an odd function; that is, sin(-x) = -sin(x).) (see Handout 4.)

 x 2x y = sin(2x) y = sin(-2x) 0o 90o 180o 270o 360o
 x 2x y = sin(4x) y = sin(-4x) 0o 90o 180o 270o 360o

4.)  Using the tables, make a conjecture about how the graphs of y = sin(-2x) and y = sin(-4x) compare to the graphs of y = sin(2x) and y = sin(4x) respectively. How will the period be affected?

5.)  Using the ExploreMath activity, check your conjectures in #4 by comparing the graphs of y = sin(2x) and y = sin(-2x) and by comparing the graphs of y = sin(4x) and y = sin(-4x).

 To make it easier to see the relationship between the graphs in #5, click on the red arrows and the magnifying glasses until the first x-intercept you see on the graph is at 0o and the last x-intercept you see is at 180o.

 Teacher Note: Some students may incorrectly assume that sin(-2x) = -2sin(x).

6.)  Take the formula you developed earlier for finding the period when b > 0 and adapt it so it can be used to find the period for R - {0}. Using a variety of b values, test the validity of your formula algebraically and using ExploreMath.