Here are some excerpts of student explanations as they reflect on how coefficients affect wave graphs and circular rotations. I pasted a different student response here for each question so you can read a variety of student "voices" as they attempt to articulate the concepts.
Student A - Question #1 asks the student to analyze the effects of coefficients on cosine wave and on the circle.
1. For the function f(x)=A*cos(Bx), A determines the radius of the circle and it therefore changes the amplitude of the cosine graph. In the circle, if A is greater than 1, for example 2, then the radius would be bigger because it would be 2 and the wave would be taller because it would go from 2 to -2. If it is less than 1, like 0.5, then the radius would be 0.5 and the wave amplitude would be smaller because it would go from 0.5 to -0.5. If the A happens to be negative then the graph would flip across the x-axis. When you flip the wave graph vertically along the x-axis the graph would start at -1 instead of 1. A horizontal flip across the y-axis wouldn’t matter for cosine.
If B is greater than 1 then the point takes less time to make a period because it is moving faster. For example, if B would be 2 then you would complete 2 periods in the time it normally takes to make 1. It B is less than 1, i.e.: 0.5, you are slower, thus it takes you twice the time to make a period. For B in the cosine wave, it doesn’t matter if it is negative or positive because it will be the same x-value regardless if the point in the circle is moving clockwise or counterclockwise.
Student B - Question #2 asks the student to analyze the effects of coefficients on sine wave and on the circle. Here, some of the explanations may duplicate part of their answer to #1, but I'm looking for the student to articulate that the sine wave flips if you make B negative, because it matters greatly whether your point rotating around the circle is going clockwise or counterclockwise, if you're tracking its height over time. This kid's response isn't perfect, but she's getting most of the ideas.
A affects the amplitude of the graph and makes the circle smaller or larger. Say A=4 will make the graph have its y-maximum at 4 and y-minimum at -4; the circle will have a radius of 4. Say A=0.5 will make the graph have its y-maximum at 0.5 and y-minimum at -0.5; the circle will have a radius of 0.5.
This means that the point along the circle will have more or less height over a certain amount of time.
B affects the wavelength distance and the speed a point moves along a circle. Say B=4 will make the wavelength shorter because it will travel more distance over a certain amount of time. B=4 will make the point in a graph move faster. B=0.5 will make the graph have a longer wavelength. B=0.5 will make a point in a graph travel much slower.
This means that the point along the circle will travel in a slower or faster pace the height in a certain amount of time.
Sine is the height of the point along the circle over time.
*Signs of B do affect the shape of the sine graph because it is not symmetrical. This means that a point in the opposite side of another point, won’t have the same height.
Student C - The next two questions, #3 and #4, are fairly straight forward. The kids need to explain how you could manipulate sine and cosine functions in order to make the circle bigger, smaller, or to turn the rotational direction around.
3. In order for you to create either a larger or smaller circle you have to change the number in front of the sine or cosine formula. If you want the circle to be bigger the number in front has to be greater than 1 and if you want the circle to be smaller the number in front has to be less than 1. For example for a circle to be smaller the number has to be less than 1 such as C=(0.25cos(t), 0.25sin(t)). This will make the circle be smaller and have a radius of 0.25. On the other hand if you want the circle to become larger the value has to be replaced by a number greater than 1 such as C=(2cos(t), 2sin(t)). This will make the circle larger by augmenting its radius to 2.
4. In order for the point of the circle to rotate in a clockwise rotation you have to change the formula by adding a (-) sign before the (t) value. An example to make the rotation clockwise is C=(cos(-t),sin(-t)).
Student D - Questions 5 and 6 ask the student to analyze a given sine or cosine equation to tell me about the circular motion that is (partially) described. I'm looking for them to describe the size of the circle, its speed, the direction of rotation along the circle, and the (x, y) point along the circle where the point starts rotating at 0 seconds.
5. m(x) = 2sin(-5x) models a circle with a radius of 2. Point C is moving in a clockwise direction, meaning that it first goes down. It’s going 5 times faster than normal, which would be 5 radians/second. Point C starts at (2,0).
Student E - See above. This kid should have shown more work or explained how they got to their starting point (x, y).
6. What behavior is modeled by n(x)= 0.5 cos(x+3)? Include in your description the size of the circle, the speed, the direction and the starting location of the point that is moving along the circle.
Size: diameter of 1 (radius of 0.5)
Speed: Starts 3 seconds “before”- has a “head start”
Starting point on the circle: (0.5, 0.7056)
Student F - For #7, the students needed to try to create a point that's rotating around a circle NOT centered at (0, 0).
7. [In order to create a circle whose center is not at (0, 0)...] I plugged in the formula, C = (cos(t)+3, sin(t)+3) and the center of the circle is (3,3). Sine would be f(x) = sin(x)+3 and cosine would be f(x) = cos(x)+3
This activity was very helpful to our later discussions on how to graph sine and cosine graphs by hand and how to set up and solve for simple trig equations, and what it all means. Stay tuned for more follow-up worksheets and discussions! :)