Philippa M. Rhodes

Write-up 10

For various a and b, invesitgate

x = a cos (t)
y = b sin (t)

A parametric curve in the plane is a pair of functions

x = f (t) and y = g (t)

where the two continuous functions define ordered pairs (x, y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t. Often, we think of x and y as varying with time (t) or as the angle of rotation that some line makes from an initial location.


First, when a = 1 and b = 1, we have a circle of radius = 1.



x = cos (t)
y = sin (t)

Now, let a = 1, and vary the value of b. We obtain the following graph when b = 3 (or b = -3).




x = cos (t)
y = 3 sin (t)

We see that our circle has been stretched into an ellipse. It still passes through points (1,0) and (-1,0), but changing b to 3 (or to -3), pulls the circle up to (0,3) and down to (0, -3).

Now, let's see what happens when b is between 0 and 1. So, let a = 1, and let b = 0.35.



x = 1.00 cos (t)
y = 0.35 sin (t)

Once again, we see that changing b gives us a graph of an ellipse that changes the points on the y-axis to (0, 0.35) and (0, - 0.35).

After several examples, it appears that as the value of b varies, the graph intesects the y-axis at points (0, b) and 0, - b). This happens because the values of t when the graph crosses the y-axis are /2 and 3/2. Since sin (t) is 1 at each of those values, we see that y = b sin (t) = b * 1 = b. Also, since cos (t) = 0 for those values of t, then x = 0.

Now, let b = 1 and vary the value of a. First, let a = 3 (or - 3).


x = 3 cos (t)
y = sin (t)

We see that this stretches the circle to the left and right to (3, 0) and (-3, 0), but the graph still passes through points (0, 1) and (0, -1).

Again, after several examples, I have concluded that changing the value of a, will change the x-intecepts to (a, 0) and (- a, 0). The values for t when the graph intersects the x- axis are 0 or 2, and . Therefore, at these points, cos (t) is 1 and sin (t) is 0. So, x = a cos (t) = a, and y = b sin (t) = 0.
Estimate the parametric equations for this graph?

How would you graph

x = .66 cos (t)
y = 13 sin (t) ,

for 0 t 2?


In conclusion, the graph of the parametric equations x = a cos (t) and y = b sin (t), for 0 t 2, will pass through points (a, 0), (- a, 0), (0, b), and (0, - b). All other points will "fit" so that the graph is symmetric about the x- axis and the y- axis.



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