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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