Every quadratic function of the form will have a y-intercept where x = 0. There are two possibilities.
The first is that the y-intercept is the turning point. If this is the case, the turning point is at (0, c). It is also well known that if the turning point is on the y-axis, that b = 0. So that would mean as well and therefore works as a formula for finding the turning point when the turning point is on the y-axis.
The second case is where the y-intercept is not the turning point. We know that the graph of a quadratic is symmetric. Since the turning point is not on the y-axis, that means there will be a second point which has the same y-value of c. If we average these x-values, we will get the axis of symmetry / x-coordinate of the turning point.
So the axis of symmetry will be the average of these two points due to the symmetry of the graph. Averaging these we find
So this means that we can always find the axis of symmetry of a quadratic of the form using .