Calculus problem with parameters.?

Question

I have stared at this problem for a while now and still nothing pops to my head. It says "Eliminate the parameter and state the endpoints of the parametric equation defined by: x=t^2-3, y=t+2, -2≤x≤2". Any help would be appreciated. Thank you.

Answer 1

Eliminate the parameter means get rid of the "t".

From the second equation: t = y - 2

And then put this in the first equation:

x = (y - 2)^2 - 3

(x + 3) = (y - 2)^2

y = SQRT(x + 3) + 2

Not sure what they mean by "state the endpoints of the parametric equation".  I take this to mean they want the value of "t". So:

-2 = t^2 - 3 ... 1 = t^2 .... and t = 1

2 = t^2 - 3 .... 5 = t^2 .... and t = SQRT(5)

So the t values for the endpoints are t = 1 and t = SQRT(5).

Or if the they want (x,y) then:

x = -2 and y = t + 2 = 1 + 2 = 3 ........ So (-2,3)

x = 2 and y = t + 2 = SQRT(5) + 2 .... So (2,SQRT(5)+2)

Answer 2

t = y - 2, so x = (y - 2)^3 - 3.

What are end points of equation?

- 2 < t^2 - 3 < 2

1 < t^2 < 5.

so 1 < t < rt 5 or - rt5 < t < -1.