In today's ever changing society, it is important to keep up with technology. Math students don't only need to know how to perform simple math operations, but they also need to know how to use computers and software to explore the unknown "math world". EMT 668 is a class that teaches the teacher how to use unfamiliar technology in the classroom to explore new mathematical concepts. This write-up is to show teachers how they can use software programs like Algebra Expresser and Theorist to explore the sine curve y = a sin (bx + c).

y = sin (x)

 Automatically, students can look at the graph of functions and you can discuss ranges, domains and special features of the sine curve. Students usually are curious and want to explore different "What if" situations. The software package, Theorist, allows you to easily explore different sine graphs.

 First, I would explain to the students that curves are easier to evaluate if you vary only one coefficient at a time. Then, I would allow the class to choose which variable, a, b, or c, they would like to explore. I chose to look at different values of a. I would tell them since we know what the a = 1 graph look like, lets graph a = -1. The following is the graph of y = (-1) sin (x):



Students can now see the effects of a = -1 on the sign function. Educators easily explain the curve begins at point (0,0) then decreases in the positive x direction. Unlike the y = sin (x) curve , that increases in the positive x direction. Now, an educator can explore other values of the variable "a".

First, I would explain to the students you can usually understand the effects of changing coefficients if you explore positive values of "a". Then, explore the same negative values of "a". Lets first look at a = 2 and a = 1/2.

y = 2 sin (x)






y = 1/2 sin (x)

The range or amplitude of the graph is 2 to -2 and 1/2 to -1/2 respectively. Students may want to explore other larger positive values of "a" . Each of these graphs gives similar results as above. Students will quickly draw conclusions about positive values of "a": The amplitude of the graph is the maximum value of y = a sin (x).

Next, an educator allows the students to explore negative values of "a". It would be helpful if we looked at "a" = -2 and - 1/2.

y = - 2 sin (x)


y = - 1/2 sin (x)

Now, students can see the maximum amplitude does not change. However, the graph decreases in the positive x direction from point (0,0) instead of increasing like the positive "a" values. Graphing the following information, student can make some generalizations: The amplitude of graphs are directly related to the "a" coefficient in the equation y = a sin (x). The amplitude is the absolute value of the maximum value of the function y = a sin (x) or educators can show the amplitude of the graph is one half the absolute value of the difference of the maximum and minimum function values. Also, if "a" is less than zero, the curve is the reflection of the graph of the function y = |a| sin (x) over the x-axis.

We need to continue to explore "What if?" questions from the students for the y = a sin (bx + c) curve. Assign "a" = 1, "c" = 0, and explore different values of "b". Educators should lead the discussion by asking students to predict graphs for different values of "b" and their reasoning behind their predictions. Start the discussion with a parent graph

y = sin (1 x)

Now that students have the parent graph fresh in their mind, ask the students which variable would they like to plug in for "b". Usually, students want to take a similar route to the one used for the "a" coefficient. First, I would try "b" = 2 and 1/2.

y = sin ((1/2)x)

y = sin ((2)x)

The graphs show the curve repeats itself twice as fast for the variable "b" = 2 and half as fast for the variable "b" = 1/2. For absolute values of "b" less than one, the graph stretches and will not repeat itself as quick. However, if the absolute value is greater than one, the curve shrinks and repeats itself quicker.

The instructor needs to encourage explorations of graphs for negative "b" values. The plot of "b"=
-2 and -1/2 looks like the following:

The graphs follow a similar pattern to the example above, y = sin (-x). Instead of the y value increasing, the y value decreases in the positive x direction for the origin. Also, the functions show the same characteristics of the positive "b" graphs above. So, what does all this mean? If the coefficient of "b" is negative, then the graph will go downward at the origin and the curve will repeat its cycle quicker for absolute values greater than one and repeat less frequent for absolute values less than one. After students completely understand the manipulations "b" makes to the graph, an educator should introduce the term, period, for this type of modification to the curve.

Now, lets explore different values of "c" for the equation y = a sin (bx + c). First, ask the students to predict what the graph would look like if you substituted "c" = 1. Remind students of the manipulations we made to the parent graph y = sin x and how the variables "a" and "b" affected the graph.

y = sin (x + 1)

It is very hard to tell exactly what happened to the graph. The plot of y = sin (x+1) seems to be off center from the parent graph. Lets continue the investigation by looking at larger values of "c" = 2, 3, and 4.

Since all the graphs are plotted on the same set of axes, students can easily see the curve shifting to the left as the value of "c" increases. Now, I would ask students to predict what the graph would do if we substituted "c" = -2, -3, and -4.

Students can see the graph has shifted to the right 2, 3 and 4 units. Educators can now introduce the term phase shift to students and the students can associate the new term phase shift to physically moving the graph in one direction or the other.

An instructor should try to bring all three terms together by transforming the parent graph step by step. First, lets display the graph the has an amplitude of 2.

y = 2 sin(x)

Now, explain to the students that the period of the functions is 360 degrees divided by "b". So, an equation that is said to have a period of 90 degrees, would actually have 4 for the value of "b". Let's graph the function that has an amplitude of 2 and the period = 4.

y = 2 sin (2x)

Notice, the height of the graph is 2 units, but the graph is repeating its cycle twice as fast as the original equation. Finally, ask the students what the graph would look like if the phase shift = 2 units to the right. In the discussion, the students should realize this would make "c" = -2.

y = 2 sin (2x -2)

By plotting the function, step by step, students can see how each step transforms the graph into a new graph.

The use of technology in the classroom allows the instructor more time to explore "what if?" situations instead of wasting time plotting tedious graphs. The students time is utilized more effectively exploring new concepts and looking at more examples of different types of graphs.

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