Assignment 10: Parametric Equations Investigations
by Shawn Broderick
I have done a few investigations with some of the problems from Assignment 10 in the Graphing Calculator program.
1. We take a look at y = cos(t), y = sin(t), t = 0 to two pi. This creates a circle as seen below in Figure 1:
Figure 1
Both sine and cosine go through their cycle from -1 to 1 and the parametric equation combines their values together.
2. When we vary cosine twice as fast, we get a parabola, as seen in Figure 2:
Figure 2
In the following movie, we make a variable n (in the equation y = cos (nt)) which cycles through the numbers 0-10. We can briefly see the circle when n = 1 and the parabola when n = 2. It is very interesting because as n goes toward 10, the figure looks to coil around itself and get tighter and tighter.
When we vary the coefficient of t in the equation y = sin (t) (i.e., y = sin (nt)) when n = 2, we get this figure-8 looking curve:
Figure 3
3. Here we are looking at when we vary a scalar in front of the cosine function in our parametric equation (see Figure 4):
Figure 4
Here we see that an ellipse is made. Notice that the major axis is the x-axis when we modify the cosine function. Also, if we multiply by 2, we can see that the major axis spans the x-axis from -2 to 2. We can deduce that similar effects will occur when the sine function is scaled. However, the major axis would be the y-axis with the sine function modifications.
4. Figure 5 below shows what happens when you graph the given equation. If you vary t from negative pi to pi, you get a nearly closed circle:
Figure 5
Figure 6 shows what happens when the top squared in the top equation is changed to the third power. Here we see that the curve is similar to the cube root of x.
Figure 6
In Figure 7 we change the top number again to the fourth power and see that it is like the inverse of x to the fourth power. Very interesting...
Figure 7
In Figures 8 and 9, we change the power of t in the denominator of the top equation and see some figures that aren't too interesting, but it is nice to see what happens.
Figure 8
Figure 9
In Figure 10, we change the coefficient of the numerator of the bottom equation. This had the same effect as changing the scalar of the sine equation from Problem 1. However, our figure is still incomplete. Also, the entire major axis is equal to the coefficient instead of twice the coefficient.
Figure 10
Changin the number to 2 made it go funky:
Figure 11
Changing the power of t in the denominator of the second equation also had a funky result:
Figure 12
As far as modifying the numbers to understand what was going on, I am not entirely sure how they work. However, in their most basic form, the equations were similar to those in Problem 1.
11. Figure 13 shows the graph of the equation given in Problem 11, with a = 1 and b = 2:
Figure 13
We now look at variations of a and b, and in the end we find a pattern. Figure 14 has a = 1 and b = 4:
Figure 14
Figure 15 has a = 2 and b = 3:
Figure 15
Figure 16 shows the curve when a = 12 and b = 13:
Figure 16
What I've gathered from this set of figures is that there is a correlation between a and b and the number of "border hits" the curve makes with the 4 by 3 rectangle, perhaps determined from the scalars on the front of the equations. For example, Figure 13 shows a curve that has 2 hits on the top and bottow borders and one hit on the sides. Figure 14 shows on hit on the sides again and four hits on the top and bottom. Therefore, we can conclude that the a determines the number of hits on the sides and the b determines the number of hits on the top and bottom. Pretty neat!
Here is a movie that varys the a and the b to see the number of hits increase and decrease on the cuve corresponding to the increases and decreases of ns. I beleive that the a and b have a differential of 1.