Re: parametric: x = cos(t), y = sin(t)

Daniel C. Bastos wrote:

> 
> Okay. I was having trouble with your notation, so I wrote x + 3 = t and
> then y = 3(x + 3) - 11 = 3x + 9 - 11 = 3x - 2. So these two are
> equivalent. They must also have the same graph. I still follow you.

The parametric equations are obviously not equivalent.

The graphs, although they may look the same after completion of a 
sketch, are not the same.  How were they traced?  Or to ask it more 
properly, what are the orientations of the graphs?

The graph of a set of parametric equations is defined as the set of 
points (x,y) obtained as the parameter varies over some interval.

Consider the same parametric equations and the interval [0,1].

a) x = t
    y = 3t - 2

For t = 0, x = 0
            y = 3(0) - 2 = -2  ...the point (0,-2)
For t = 1, x = 1
            y = 3(1) - 2 = 3 - 2 = 1   ...the point (1,1)

b) x = t - 3
    y = 3t - 11

For t = 0, x = 0 - 3 = -3
            y = 3(0) - 11 = -11  ...the point (-3,-11)
For t = 1, x = 1 - 3 = -2
            y = 3(1) - 11 = 3 - 11 = -8   ...the point (-2,-8)

Note that all four points satisfy the rectangular equation y = 3x - 2. 
Now reconsider what were were discussing about the equivalence relation 
between 360deg, and -360deg.  Specifically, think of how this applies to 
a circle traced clockwise from a certain point vs. the same circle 
traced counterclockwise from the same point.

-- 
Darrell