The vertex is pretty easy. We just need to complete the square .
First, the leading coefficient of the polynomial must be 1, so we need to factor the -1. That leaves us with y=-1(x^2-8x-2). Now, the purpose of completing the square is to find a constant that makes x^2-8x a perfect square. To that, we use this formula: c=(1/2*b)^2 or (1/2*8)^2, which is 16.
Now we know that we have to add 16 to make it a pefect square, but because we cannot just add something on one side of the equation, we need to "get rid of it" too. We could add 16 on both sides, or we can just add 16 and then subtract it immediately, which is the same thing. Either way works :)
y=-1(x^2-8xcolor(green)(+16-16)-2)
y=-((x^2-8x+16)-16--2)
x^2-8x+16 is a perfect square, so let's symplify it
y=-((x-4)^2-16-2)
y=-((x-4)^2-18)
Now we just distribute the negative:
y=-(x-4)^2+18
The equation is now in vertex form.
It's easy to find the vertex from this point:
y=-(x-color(red)(4))^2+color(purple)(18)
(color(red)(4), color(purple)(18)).
Finding the x-interecpt means setting y=0 and solving for x:
0=-(x-4)^2+18
-18=-(x-4)^2
18=(x-4)^2
+-sqrt(18)=x-4
4+-sqrt(18)=x
or x=4+-3sqrt2
Those are the exact values. If you want the estimated values, they're x~~8.243 and x~~-.0.243
To find the y-intercept we just set x=0 and solve for y:
y=-(0-4)^2+18
y=-(-4)^2+18
y=-16+18
y=2
