Question:

Calculus problem with parameters.?

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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.

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  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)


  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.  

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