# How can I calculate the vertex of a parabola?

Jul 21, 2014

If parabola is defined by a quadratic function $y = a {x}^{2} + b x + c$, it's vertex is a point of extrema (minimum for $a > 0$ and maximum for $a < 0$).

Therefore, the easiest way to determine this point is to take the first derivative from a given quadratic function and equate it to zero. The solution of this equation gives an $x$-coordinate of a vertex. The $y$-coordinate can be obtained by substituting the just found $x$-coordinate into a given quadratic function.

The first derivative of a function $y = a {x}^{2} + b x + c$ is $2 a x + b$. Therefore, we have to solve the following equation:
$2 a x + b = 0$

Solution:
$x = - \frac{b}{2 a}$

Substituting this into a given quadratic function to get $y$-coordinate:
$y = a {\left(- \frac{b}{2 a}\right)}^{2} + b \left(- \frac{b}{2 a}\right) + c = {b}^{2} / \left(4 a\right) - {b}^{2} / \left(2 a\right) + c = \frac{4 a c - {b}^{2}}{4 a}$

Therefore, the $\left(x , y\right)$ coordinates of a vertex of this parabola are
$\left(- \frac{b}{2 a} , \frac{4 a c - {b}^{2}}{4 a}\right)$