# How do you find the vertex of a parabola without graphing?

Apr 14, 2015

Assuming the parabola has not been rotated
the vertex of the parabola occurs where the derivative of its function is equal to zero.

Determine the general form of the derivative (this should be a linear function in terms of $x$)

Set the general form of the derivative to zero and solve for $x$

Substitute the value you obtained for $x$ back into the expression for the parabola to get the $y$ component of the vertex coordinate.

Example: If the parabola is defined by $y = f \left(x\right) = 3 {x}^{2} + 12 x + 7$

$f ' \left(x\right) = 6 x + 12$

the vertex occurs where
$6 x + 12 = 0$
$\rightarrow x = - 2$

$y = 3 {\left(- 2\right)}^{2} + 12 \left(- 2\right) + 7$
$= - 5$

The vertex is at $\left(- 2 , - 5\right)$