The Vertex of a Parabola

The vertex of a parabola is the point where the parabola crosses its axis of symmetry.   If the coefficient of the x2 term is positive, the vertex will be the lowest point on the graph, the point at the bottom of the “U”-shape.  If the coefficient of the x2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “U”-shape.

The standard equation of a parabola is

y = ax2 + bx + c.

But the equation for a parabola can also be written in "vertex form":

y = a(xh)2 + k

In this equation, the vertex of the parabola is the point (h, k).

You can see how this relates to the standard equation by multiplying it out:

y = a(xh)(xh) + k

y = ax2 – 2ahx + ah2 + k

The coefficient of x here is 2ah. This means that in the standard form, y = ax2 + bx + c, the expression

gives the x-coordinate of the vertex.

Example:

Find the vertex of the parabola.

y = 3x2 + 12x – 12

Here, a = 3 and b = 12. So, the x-coordinate of the vertex is:

Substituting in the original equation to get the y-coordinate, we get:

y = 3(–2)2 + 12(–2) – 12

= –24

So, the vertex of the parabola is at (2, 24).