By completing the square, a quadratic in the form ax2+bx+c can be written in the form

for some h,k∈R.

Notice that the minimum / maximum (depending on a) of this expression occurs when (x−h)2=0, ie x=h. The resulting y-coordinate is then simply k. Thus the coordinates of the vertex are (h,k). We call this form the vertex height form as it illustrates the connection between the quadratic and its vertex.

Another way to find the coordinates of the vertex is to remember that it lies on the axis of symmetry, so its x-coordinate is −2ab​. By plugging this value into ax2+bx+c, we can find the y-coordinate

Example

Find an equation for the parabola with vertex (3,4) passing through the point (0,−14).

In vertex height form, the parabola has equation

And since (0,−14) lies on the parabola:

So the equation is −2(x−3)2+4.