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* = *ax*^{2} + *bx* + *c*.

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

*y* = *a*(*x* – *h*)^{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*(*x* – *h*)(*x* – *h*) + *k*

*y* = *ax*^{2} – 2*ahx* + *ah*^{2} + *k*

The coefficient of *x* here is **–**2*ah*. This means that in the standard form, *y* = *ax*^{2} + *bx* + *c*, the expression

gives the *x*-coordinate of the vertex*.*

**Example: **

Find the vertex of the parabola.

*y* = 3*x*^{2} + 12*x* – 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).