Change parabolic equation to canonical form

Question

I have equation $y = -x^2 + 2x + 7$. How can I change it to canonical form, which looks like $y^2 = 2px$ ? ($p$ will be parameter)

What i ve tried so far: $$\begin{align} y &= -x^2 + 2x + 7\\ y &= -(x^2 - 2x + 1) + 8\\ (y-8) &= -(x-1)^2 \\ (y-8)^2 &= 2*(0.5)*(x-1)^4 \end{align} $$

But I have read somewhere its wrong, so how do I make it correct?

Or is my solution correct?

Answer 1

$$-y=x^2-2x-7=(x-1)^2-8\implies (x-1)^2=-(y-8)$$ which is of the form $(x-\alpha)^2=-4a(y-\beta)$

Now, if we are allowed to make the transformation of axes, we can set $x-1=Y,y-8=X$

So,$Y^2=-X=2\left(-\frac12\right)X$

Answer 2

$$y = -x^2 + 2x + 7 $$ $$y = -(x^2 - 2x +1)+8 $$ $$y = -(x- 1)^2+8 $$ $$(x- 1)^2=-(y-8) $$ $$(x- 1)^2=2(-\frac{1}{2})(y-8)\Rightarrow p=-\frac{1}{2},x-1=Y,y-8=X $$ $$Y^2=2pX$$