Another question regarding parametric equations (screenshot below about what I'm confused about in a moment). Is it impossible for two parametrized equations with identical non-parametrized equations to be really seen apart by their non-parametrized equation formulas? (ex. below in a sec.)

Question

mendicant_bias · Answer

Let's say I have two parametrized functions, one where$$x = \sqrt{t}, y = t$$
another where$$x = t, y = t^{2}$$
If I put these both in non-parametrized from, they would be identical;$$y = t, x = \sqrt{t}, y = t = x^{2}$$$$y = t^{2}, x = t, y = t^{2} = x^{2}$$
$$1.) y = x^{2}$$$$2.) y = x^{2}$$

However, if the parametrized functions are actually plugged in for values of t, you'll obviously get different results. Isn't there something a little inherently flawed/wonky about this? Their parametrized forms are non-identical and give different results, but their un-parametrized forms are.

mendicant_bias · Answer

(This isn't really a problem, so much as it seeming really flawed or crazy that you can end up with these things equalling the same thing. If I had a non-parametrized equation and put it into a parametric form, how would I know which parametric form is "right"?

anonymous · Answer

you can convert a non parametrized equation in to different parametric forms. All forms would be correct but t or the parameter isnt equal. so if t=1 in first form and t= 1 in second form would be located on the same curve but not in the same point