For this question:
Parameterize $y^2 - x = 5.$
My solution was
$x(t) = t$
$y(t) = \sqrt{5+t}$
However, when I checked with ChatGPT (because there are no solutions for the textbook we are using in my multivariable calculus course,) it stated that this only captured the positive half of the parabola. I suggested adding a +/- symbol, but instead, it offered this as a suggestion to capture both halves:
For 2A, your suggestion is indeed another way to capture the full parabola in a single equation. However, it's important to note that the convention is to use piecewise-defined functions when the expression inside the square root can be negative. By writing y(t) = +/- sqrt(5+t), you're implying that the function takes both the positive and negative square root at every value of t. That's not the case here. For t > -5, you have two real roots (one positive, one negative). For t = -5, you have one root (0), and for t < -5, you have no real roots. So, it's more precise to express the solution as a piecewise function:
y(t) = sqrt(5+t) for t >= -5 (the upper half of the parabola)
y(t) = -sqrt(5+t) for t >= -5 (the lower half of the parabola)
This explicitly shows that the positive and negative roots are separate.
Its exact answer seems contradictory but the idea seems right. Would anyone be able to help me with the solution that captures both halves?
Since the y is squared, avoid the square root altogether.