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 x 2 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 x 2 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=a x 2 +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=a x 2 2ahx+a h 2 +k .

This means that in the standard form, y=a x 2 +bx+c , the expression b 2a gives the x -coordinate of the vertex.

Example:

Find the vertex of the parabola.

y=3 x 2 +12x12

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

12 2(3) =2

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) .