Exploring Period...Period.

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.

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

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 = 0^o, and d = 0.  What does this graph look like when b = 1?

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?

4.)  Click on the red circle if necessary and the magnifying glasses to make -360^o the first labeled tic mark on the x-axis and 360^o 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.

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