# How do you find the vertex of a parabola using the axis of symmetry?

Apr 11, 2015

Knowing the axis of symmetry is not enough to determine the vertex.

The axis of symmetry contains infinitely many point, one of which is the vertex of the parabola. Without more information it is impossible to determine which of those infinitely many points is the vertex.

Apr 11, 2015

When you're given the quadratic equation of the parabola, you can find it's vertex using the axis of symmetry.

The axis of symmetry in a quadratic equation would always be $x = - \frac{b}{2 a}$.

Here is an example of a quadratic equation
$y = {x}^{2} - x + 3$

We can find the axis of symmetry by using $x = - \frac{b}{2 a}$.
The axis of symmetry in this case would be $x = - \frac{- 1}{2 \times 1} = \frac{1}{2}$

We can now find the y=coordinate of the vertex of the parabola by substituting $x = \frac{1}{2}$ into the quadratic equation of $y = {x}^{2} - x + 3$.

Sub $x = \frac{1}{2}$ into $y = {x}^{2} - x + 3$

$y = {\left(\frac{1}{2}\right)}^{2} - \frac{1}{2} + 3$
$y = \frac{1}{4} - \frac{1}{2} + 3$
$y = - \frac{1}{4} + 3$
$y = 2 \frac{3}{4}$
Hence, the vertex or turning point of $y = {x}^{2} - x + 3$ is $\left(\frac{1}{2} , 2 \frac{3}{4}\right)$ graph{x^2-x+3 [-4.83, 5.17, -0.4, 4.6]}