PARAMETRIC EQUATIONS: COMPONENTS

Change the parameters of the parametric equations of the red ball below and click "(re)start".

x(t) =
y(t) =

How are the three motions related?
         
(Re)start the animation several times ... Use the "pause" button. Make sure you puzzle out what is going on here!

The animation shows three graphs: The red y-x graph (y is a function of x), and its component x-t and y-t graphs (they are functions of t).

Qualitatively, you should notice that the co-ordninates of points of the red graph correspond exactly to the movements on the X-axis and Y-axis. The red graph consits of all the (x, y) pairs generated by the parametric x(t) and y(t) equations.

Quantitavily, you should notice the x-y values. Record the co-ordinates after 1 second, 2 seconds, ... Can you deduce the cartesian (algebraic) equation of the red graph, i.e. y as a function of x from the values?
Can you deduce the cartesian equation from the two parametric equations of the red ball? How is this related to the gradient of the graph?

The gradient of the red graph =    

Now check your conjectures by changing the parametric equations, e.g. x(t) = 8t and y(t) = 12t. Click "(re)start".

Now change the equations to linear form, e.g. x(t) = 10 + 8t and y(t) = 5 + 12t. Investigate other cases and make a conjecture. Can you deduce the cartesian equations? Describe your method ...

Try some quadratic and trigonometric equations ...