how to find vertex of parabola

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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

$y = a x^{2} + b x + c$ .

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:

$\begin{array}{l} y = a (x - h) (x - h) + k \\ y = a x^{2} - 2 a h x + a h^{2} + k \end{array}$ .

This means that in the standard form, $y = a x^{2} + b x + c$ , the expression $- \frac{b}{2 a}$ 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:

$- \frac{12}{2 (3)} = - 2$

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

$\begin{array}{l} y = 3 {(- 2)}^{2} + 12 (- 2) - 12 \\ = - 24 \end{array}$

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