Parametric equation of a shifted parabola

Question

How to write the parametric equation of shifted parabola? For example, I thought about the equation of a parabola with $a=1$ and vertex being $(3,2)$ would be $((t-3)^2,2(t-2))$, but it is $(t^2 +3,2t+2)$.

Answer 1

You're confusing parametric equations with implicit equations; you wrote $t-3$ instead of $x-3$. The unshifted parabola is

$$x = t^2$$ $$y = 2t$$

so the shifted parabola is

$$x-3 = t^2$$ $$y-2 = 2t$$

or

$$x = t^2+3$$ $$y = 2t+2$$

or

$$(x,y) = (t^2+3,2t+2)$$

Answer 2

The equation of parabola in Cartesian coordinates is $$x=(1/4)(y-2)^2 +3 $$

For a simple parametrization we choose $t=(y-2)/2$ which gives us $x=t^2 +3$ and $y=2t+2$ as it was suggested by the answer.

Note that parametrization is not unique and one should check the answer carefully.