 # Vertex Formula

In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

For example, a square has four corners, each corner is called a vertex. The plural form of the vertex is vertices. The word vertex is most commonly used to denote the corners of a polygon.

When two lines meet at a vertex, they form an included angle. For polygons, the included angle at each vertex is an interior angle of the polygon. Vertex is also sometimes used to indicate the ‘top’ or high point of something, such as the vertex of an isosceles triangle, which is the ‘top’ corner opposite to its base, but this is not its strict mathematical definition. The Vertex Formula is given as,

$\large Vertex=\left(h,\:k\right)=\left(\frac{-b}{2a},c-\frac{b^{2}}{4a}\right)$

### Solved Examples of Vertex

Example: Find the vertex of the parabola: $y = 3x^{2} + 12x – 12$

Solution:

Given,
a = 3
b = 12

c = -12

So, the x-coordinate of the vertex is:

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

y-coordinate is:

(4ac – b2)/4a = [4(3)(-12) – (12)2]/ 4(3)

= (-144 – 144)/12

= -228/12

= -24

Alternatively,

Substitute x = -2 in the original equation to get the y-coordinate as given below:

y = 3(-2)2 + 12(-2) – 12

= 12 – 24 – 12
=-24

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