Question

the vertex of the parabola below is at the points (4,-1) which of the equations below could be the one for this parabola

Answer 1

The right answer is C.

$x=-2(y+1)^2+4$

This parabola open left. Recall that a parabola is a curve where any point is at an equal distance from a fixed point called the focus and a fixed straight line called the directrix. So, the standard form of the equation for parabolas opening left (or rigth) is:

$(y-k)^2=4p(x-h)$

where $p$ is the distance from the focus to the vertex and is the same as the the distance from the vertex to the directrix.

As in this problem the parabola opens left then it is true that $p<0$. To prove the truth of this let's substitute the vertex in the equation like this:

$V(4,-1) \\ \\ x=-2(y+1)^2+4 \\ 4=-2(-1+1)^2+4 \rightarrow \boxed{4=4} \ Proved!$