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