Vertex as a part of mathematical vocabulary has many definitions attached to it.  It is defined according to the context or topic and also often understood intuitively. Vertex generally denotes a point where you notice a converging or diverging phenomenon. The plural of vertex is vertices.
Let us see how vertex is defined in various geometrical contexts. the formulas or equations related to it and other geometrical entities derived from it.

"Vertex"  plays an important role in many geometrical situations.

Vertex of an angle
The vertex of an angle is the common end point of the two sides of the angle where the angle is formed.
The angle is also commonly denoted by the upper case letter used to indicate the vertex.

In the adjoining diagram the angle is formed
at the common  end point A,  of the sides AB
and AC. A is called the vertex of the angle BAC,
and the angle is also denoted as ∠A.
 Vertex of an Angle

Vertex in a Polygon
The vertex of each angle in a polygon is a vertex of the polygon.
In general if the polygon is of 'n' sides, then there are n angles and hence n vertices.

Pentagon ABCDE has five vertices
A, B, C, D and E as shown.
 Vertex Polygon


Vertex in a 3-D shape
Vertex in a 3-D shape like a prism is the point of convergence of edges.  

In the rectangular prism shown the eight
vertices are A, B, C, D, E, F, G and H
each formed at the intersection of three
edges of the prism.
 Vertex Prism
A Pyramid with a n sided polygon as base will have n + 1 vertices. The vertex opposite to a face which is considered as the base of the pyramid is called the Apex.
It is true that all angles of a polygon are formed at the vertices. But the term Vertex Angle is used in certain contexts.
If in a triangle any side is considered as the base, then the angle formed at the vertex opposite is called the vertex angle.
The vertex angle in an isosceles triangle is the angle included between the congruent sides.

In triangle ABC,  AB ≅ AC.  
The angle formed in between
these two congruent sides ∠A is
the vertex angle of the triangle.
It is opposite to the third side BC
which is called the base of the
triangle.
 Vertex Angle Isosceles

Example:
ABC is an isosceles triangle with AB = AC. If the measure of one of the base angles m ∠B = 65º,
find the measure of the vertex angle A.

Vertex Angle Example

m ∠B = m ∠C  = 65º                Base angles are congruent in an isosceles triangle.
m ∠A = 180 - ( m ∠B + m ∠C)
         = 180 - (65 + 65) = 50º.
Hence the measure of the vertex angle = 50º

Vertex angles in a regular polygon
A circle can be drawn circumscribing the sides of a regular polygon. The vertex angles are formed at the
circumcenter by the sides of the polygon. 

In the adjoining diagram, ABCDEF is a
regular Hexagon. The angles marked at
the circumcenter 1, 2, 3, 4, 5 and 6 vertex angles
and each angle measures = 60º.
 Vertex Angle Hexagon

There are n congruent isosceles triangles formed with vertex at the center.Thus the Vertex angles of a regular polygon are congruent.
Example:
Find the measure of each of the vertex angles of a regular Pentagon.

Vertex Angle Polygon Example

The measure of an interior angle of a regular polygon of n sides is given by the formula

$\frac{(n-2)180}{n}$

Hence the measure of an interior angle of a regular pentagon (5 sides) = $\frac{3\times 180}{5}$ = 108º.

Measure of each base angle of the isosceles triangle with vertex at the circumcenter = $\frac{108}{2}$ = 54º.

Measure of the vertex angle of a regular pentagon = 180 - (54 + 54) = 72º.

Alternate Method:
The vertex angles of a regular polygon are congruent.
Since a regular pentagon has 5 vertex angles and angle around the center = 360º,
Measure of each vertex angle = $\frac{360}{5}$ = 72º
The vertex of a parabola is the point where the axis of symmetry intersects a parabola. The Vertex of the parabola corresponds to the maximum or minimum value of the quadratic function which the parabola represents.

Examples
An upward parabola x2 = y is shown here.
The axis of symmetry is the y axis (x = 0)
and it cuts the parabola at the vertex (0, 0).
Vertex here represents the minimum function value.
 Parabola Up
Parabola Down A downward parabola (x - 2)2 = -(y - 5) is shown
here. The axis of symmetry is the vertical line
x =2 which cuts the parabola at the vertex (2, 5).
Vertex for this graph represents the maximum
function value.
 A right open parabola (y - 2)2 = 4(x - 3) is shown here.
The axis of symmetry is the horizontal line y = 2 which
cuts the parabola at the vertex (3, 2).
 Parabola Right
 Parabola Left A left open parabola y = -2(x - 4) is shown here.
The axis of symmetry is the x axis (y = 0) which
cuts the parabola at the vertex (4, 0)

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The vertex form of equation to a parabola, not only shows the vertex, but also the line of symmetry.
If the equation is of the form
(y - k)2 = 4p(x -h), the parabola is symmetrical about the horizontal axis, y =k
The parabola opens to the right if p is positive and opens to the left if p is negative.

When the equation is of the form
(x - h)2 = 4p(y - k) , the parabola is symmetrical about the vertical axis x = h.
The parabola opens up if p is positive and opens down if p is negative.

If the equation of the parabola is given in the standard form, then the vertex can be found out by observing the equation. The vertex of the parabola is given by (h, k).
The equation of the parabola has to be rewritten in standard form by completing the squares if necessary.

Solved Example

Question: Find the vertex of the parabola 2x2 - 8x + y + 6 = 0, rewriting it in standard form.
Solution:
 
2x2 - 8x + y + 6 = 0 
     2x2 - 8x = -y - 6                                 x and y terms separated
     2(x2 - 8x) = -(y + 6)

       x2 - 8x = -$\frac{1}{2}$(y + 6)   
   
     x2 - 8x + 16 = -$\frac{1}{2}$(y + 6) + 16        Completed the square on the left side

     (x - 4)2 = -$\frac{1}{2}$(y - 26)                     Equation in vertex form.

     Hence the vertex of the parabola is (4, 26). The axis of symmetry is x = 4.

     The parabola opens down as p = -$\frac{1}{8}$ is negative.