The graph of f(x) = ax^2 + bx + c is a parabola if a does not equal zero. The effect of a and c on the graph are fairly simple, a can flip it upside down and make it tighter or wider, c moves the apex in the direction of the y-axis.

b is less obvious. It would seem to move the apex of the parabola along a curve, but what curve?

Originally posted by talzamir The graph of f(x) = ax^2 + bx + c is a parabola if a does not equal zero. The effect of a and c on the graph are fairly simple, a can flip it upside down and make it tighter or wider, c moves the apex in the direction of the y-axis.

b is less obvious. It would seem to move the apex of the parabola along a curve, but what curve?

Putting it into a more familiar format of y(x) instead of the paired equations of y(b) and x(b).

Coordinates of the apex being, as you solved it,

x = -b/(2a)
y = -b^2/(4a) + c

Squaring the first equation gives x^2 = b^2/(4a^2).

Tweaking the second equation and then substituting x into it gives

y = -b^2/(4a) + c = -a * b^2 / (4a^2) + c = -ax^2 + c

So it seems not just any parabola, but identical in shape with the original, upside down, and has the apex at (0,c) which is where all the parabolas y = ax^2 + bx + c pass through the y-axis.