Question about parametric equations

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1

To demonstrate my question in an example, please watch this video (skip to 18:00, example b)

https://www.youtube.com/watch?v=nprlKjutEuk

I don’t know how to graph parametric equations. Sometimes, the graph of the parametric equation and the equation of y in terms of x differ. Like in example b: t cannot be less than -5… otherwise, the x would be undefined. at -5, x = 0. So, the graph is half a parabola with a domain of 0 or greater. (the right half. My graphing calculator shows the same, and by the way, my t min = -10.)

But if we write it as y in terms of x, we see a complete parabola with a domain consisting of all real numbers. (as shown in the video)

I’m not sure which graph to follow. Thanks for any help!

2

My question, more concisely (I think): should we graph before or after eliminating the parameter?

3

@BethanyD it should look like the parabola whose domain is x ≥ 0, i.e. only the “right” half of the parabola should be drawn. This is assuming t can be any real number.

A common error occurs when squaring both sides of an equation involving square roots (note that the square root of a real number is nonnegative by definition). When we have x = sqrt(t+5), this implies that x^2 = t + 5 <==> t = x^2 - 5. But the reverse isn’t true - that is, if t = x^2 - 5, then it is not necessarily true that x = sqrt(t+5).

4

Thank you SO much.