Assignment 10 Write-up

Billy Poe

1. Graph x = cos (t) and y = sin (t) for 0 < t < 2pi

How can we alter this graph in order to instigate a high school class discussion?

First of all, let's notice what the original graph is--the unit circle. This makes since because as t goes from 0 to 2pi, the cos and sin of t makes up a specific point on the unit circle. Notice also that the unit circle is centered at the origin.

Now, let's shake it up a bit.

2. x = cos (t) and y = 0 for 0 < t < 2pi

If you notice, the graph is a line along the x-axis from x = -1 to 1. This makes since because the minimum value that x can have is -1 since cos (pi) = -1 and the maximum value is 1 since cos (0) = 1. But why does it lie on the x-axis? Because the y-value is 0; it never changes from 0 < t < 2pi.

3. x = cos (2t) and y = sin (2t) for 0 < t < 2pi

Some students may have thought that this circle would be larger than the first, however it's not. Why? Because for this particular equation, x and y can only lie between -1 and 1. Consider the graphs y = cos (x) and y = sin (x). In each of these graphs, the x-value corresponds to the t-value in our given equations. In these graphs, it doesn't matter how great or small x is, the y-value always lies between -1 and 1, just like in our given equations.

4. x = 2cos (t) and y = 2sin (t) for 0 < t < 2pi

Now we get the graph that some of the students had probably expected earleier. Why is it bigger now? Because we are basically taking the x and y values that we got from our first demo and doubling them. So both x and y range from -2 to 2 (since they originally ranged from -1 to 1).

5. x = cos (t) + 1 and y = sin (t) for 0 < t < 2pi

This graph makes sense because you take every possible x-value and add 1 to it, so it shifts the whole circle over one unit. Allow your students to guess what would happen if it were x = cos (t) - 1 or y = sin (t) + 1.

6. x = cos (t) and y = {sin (t)}/2 for 0 < t < 2pi

This graph also makes sense because we are taking every y-value from the first demo and dividing it by half. So now the min and max values are -1/2 and 1/2. Notice that the x-values stay the same.

There are many other ways to alter this graph. Psyche your kids out with this one!