Parametric Equations

Given a function F which consists of X and Y parametric equations, each expressed as a function of the same independent variable t.
In general it is difficult for the students to imagine the graph of F.
Using the Applet of Parametric Equations, the graph of F can display and the visual explanation for the table of the derivatives of X and Y can be given.

Using the applet of Parametric Equations

In this applet you can find the locus of F, dragging the point A.
The button of　"Init"　is for replacing the figure in the initial state.
The button of　"Clear"　is for clearing the locus of F.
In the picture on the lower left, the ratios of p and q to 1 are displayed.
Where p and q are the circular frequency of X and Y respectively.
If you drag the yellow and green points, p and q will be changed respectively.
In the picture on the right, if you drag the yellow and green circles, the radii of rx and ry will be changed respectively.

Applet of Parametric Equations

Explore

Given parametric equations

To imagine the funation F, a table of derivatives of X and Y is often used.
What is the reason we use it?
The table of the derivatives of X and Y is illustlated below.

Using this applet, you can see movements of X and Y.

Now take notice of X.
If t changes from 0 to PI, X moves in the left direction.
If t changes from PI to 2*PI, X moves in the right direction.
With respect to the table above, the derivative of X is greater than 0, when X moves in the left direction.
And the derivative of X is less than 0, when X moves in the right direction.
Therefore if you know whether the derivative of X is greater than 0 or not, you will find a movement of X.

That is true of Y.
The derivative of Y is greater than 0, when Y moves in the upper direction.
The derivative of Y is less than 0, when Y moves in the lower direction.

Therefore if you know the derivatives of X and Y, it will be easy to imagine a movements of F.
For example, if the derivatives of X and Y are more than 0, F moves in the upper left direction.