Summer Tuggle

Assignment
10

9.
Investigate each of the following for

Describe each when **a
**= **b**, **a **< **b**, and **a **> **b**.

The
first pair of equations

x
= a cos (t)

y
= b sin (t)

results
in the following graph:

The
purple circle is formed when a and b are equal.

x
= 3 cos (t)

y
= 3 sin (t)

The
red ellipse is formed when b is greater than a.

x
= 3 cos (t)

y
= 5 sin (t)

It
makes sense that a larger b would elongate the original circle on the
y-axis.

When
a is larger than b, the original circle is elongated horizontally.

x
= 5 cos (t)

y
= 3 sin (t)

The
second pair of equations

x
= a (cos (t))^{2}

y
= b (sin (t))^{2}

give
this series of graphs:

The
purple line segment results when a and are both 3. The red segment shows that b is 5 and a is 3,
thus a line segment with a steeper negative slope. The blue line results when b is 3 and a
larger a value of 5.

x
= a (cos (t))^{3}

y
= b (sin (t))^{3}

gives
this series of graphs:

x
= a (cos (t))^{4}

y
= b (sin (t))^{4}

gives
this series of graphs:

x
= a (cos (t))^{5}

y
= b (sin (t))^{5}

gives
this series of graphs:

The graphs will continue to follow this cycle where
functions raised to an even power will produce a curve in the first quadrant
that continues to get more and more concave while these functions raised to an
odd power create a star shaped figure that will continue to stretch along the
x-axis.