ok so you know the two x-intercepts are given by
,i.e.
and
. Each of the x-intercepts are the same distance away (along the x-axis) from the turning point. So if you add then together and divide by 2, you should get the x coordinate of the turning point.
The reason it gives the middle point is because parabolas are symmetrical on the y-axis around the turning point, i.e. the left and right sides of the curve are the same, but flipped.
as for the y coordinate
your equation of your parabola is y=ax
2+bx+c
so sub it in
which just gives c.
Another way of looking at the y-coordinate is that for it to be a y-intercept, x must be 0. So that means that no matter what the coeffcients of x^2 and x are, they will equal 0 leaving just c.
Or look at it this way. You have your graph of y=ax
2+bx=x(ax+b)
so there will be an intercept at x and -b/a
if you shift the graph up c units, there will be a y-intecept (this is previously the point (0,0)) at (0,c)