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.