__Assignment
#10
__

__Parametric
Curves__

As we look
at parametric equations and parametric curves, we explore the following
equations and possible transformations using Graphing Calculator 3.5.

When
graphing these equations in Graphing Calculator 3.5, we use the 2-vector
function, so the equations entered in the program shows up as:

Graphing
Calculator automatically assumes the range of t is from 0 to 1 and the
following graph is shown:

We can
conveniently change the range by changing the equation to:

Increasing
the range results in the following graph:

By using the
above equation and graph as a parent function, we can now look at various
transformations using the 2-vector format to explore other graphs:

The above
equations show an addition and subtraction outside of the trigonometric
functions. The addition of 1 to the x
equation results in the circle shifting 1 unit in the positive direction along
the x-axis, while the subtraction of 2 to the y equation results in the circle
shifting 2 units in the negative direction along the y-axis.

What if we
add a number inside the trigonometric function?

By adding 1
inside the cosine function, the circle is squeezed into the following shape:

Let’s add 1
to the sine function:

By adding 1
inside the sine function, the circle is squeezed in the opposite direction:

By
increasing the addend, the circle is continually squeezed and only returns to
its normal circular shape when the addend is equal to pi or a multiple of pi.

What happens
when we multiply one of the functions by a number?

By
multiplying the x-equation by 2, the circle now takes on an elliptical shape
with the distance from the center to each vertex on the x-axis being 2.

By
multiplying the y-equation by 3, the circle takes on an elliptical shape with
the distance from the center to each vertex on the y-axis being 3