This write up is for problems #9 in Assignment 10.

Parametric Curves

by

Brad Simmons

To begin this exploration of parametric equations we will look at the graph of

x = cos(t)

y = sin(t)

for 0 < t <

Now if we let the coefficient of t vary, then we can see how that will effect the graph.

The graph below is for

x = cos(2t)

y = sin(t)

for 0 < t <

Now compare the graph for

x = cos(5t)

y = sin(t)

for 0 < t <

To continue this exploration of these parametric curves as the coefficient of t varies in the cosine function please click here for a QuickTime movie.

How will the graph change if we vary the coefficient of t in the sine function?

The graph below is for

x = cos(t)

y = sin(2t)

for 0 < t <

Now compare the graph for

x = cos(t)

y = sin(5t)

for 0 < t <

 

How will the parametric curve change if we vary the constant by which the cosine or sine function is multiplied? The parametric equations of the curve can be written as follows ...

x = a cos(t)

y = b sin(t)

for 0 < t <

The graph below is for

x = 2 cos(t)

y = 7 sin(t)

for 0 < t <

Now compare the graph for

x = 5 cos(t)

y = 7 sin(t)

for 0 < t <

 

To continue this exploration of these parametric curves as the constant "a" varies in the cosine function please click here for a QuickTime movie.

How will the graph change if we vary the constant "b" in the sine function?

The graph below is for

x = 5 cos(t)

y = 3 sin(t)

for 0 < t <

Now compare the graph for

x = 5 cos(t)

y = 8 sin(t)

for 0 < t <

If a > b, then the curve is an ellipse with a horizontal major axis. If a < b, then the curve is an ellipse with a vertical major axis. If a = b, then the curve is a circle with a radius equal to a (or b).

Now we will investigate how the curves change if each function is squared. Consider the following equations.

x =

y =

for 0 < t <

The graph below is for the equations

x =

y =

for 0 < t <

 

Now compare the graph when the equations are changed to

x =

y =

for 0 < t <

 

The graph of the curve is a line segment with endpoints ( 0, ) and ( , 0 ).

Now consider the equations if the constants a and b are not squared. What if the equations are changed to ...

x =

y =

for 0 < t <

The graph shown below is for a = 4 and b = 6.

The graph of the curve is a line segment with endpoints at ( 0, b ) and ( a, 0 ).

To take this investigation one step farther we will look at how the curves change if each function is cubed. Consider the following equations.

x =

y =

for 0 < t <

The graph shown below is for a = 7 and b = 5.

 

 

Now consider the graph when a = 4 and b = 9.

 

The graph of the curve is a "diamond like" curve with the four points of the "diamond" located at ( a, 0 ), ( -a, 0 ), ( 0, b ), and ( 0, -b).

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