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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.