What translating a vertex does to a function in completed square form

Question

I just wanted to see if I was thinking about this in the right way

Say we have the function $f(x)=2x^2-4x-7$

In compledted square form that gives us $f(x)=2(x-1)^2-9$

this gives $(1,-9)$ as the coordinates of the vertex

now lets say we have to translate that vertex to the point $(-1,-5)$

so we're mapping all x values by negative two and all y values by 4.

so then would our function become

$y+4=2(x-2)^2-4(x-2)-7$

giving us a copmpleted square form of

$f(x)=2(x-3)^2-13$

is that the right way to be thinking about this ?

Answer 1

Clearly, the answer is $y = 2(x+1)^2-5$.

Now, to correct your mistake.

It should be

\begin{align}y\color{red}-4&=2((x\color{red}+2)-1)^2-9 \\ y&=2(x+1)^2-9+4 \\ &=2(x+1)^2-5\end{align}

A possible relevant read on why we replace $x$ by $x+2$ to shift left by $2$ units.