What is a Vertex in Geometry?

When two or more lines are met at a point to form an angle, that point can be called a vertex. So vertex can be defined as a point when two or more lines meet to form an angle. A vertex is denoted by capital letters like A, B, E etc. In geometry there are many shapes like cube, square, triangle etc. For these figures, there are more than one vertex. So the plural form of a vertex is called vertices. Let’s look at a few figures

There is an Euler’s formula to calculate how many vertices are present for a three dimensional (3D) figure. The formula is given by-

Euler’s Formula-> F + V – E = 2

The above formula can be modified to get vertices count as

V = E + 2 – F

Where

V represents number of vertices

F represents number of faces

E represents number of edges

Let’s look at the few questions on finding the number of vertices for the given figures.

Question 1: Find the number of vertices present for a figure (cube) with 6 faces and 12 edges.

Solution:

Given

Number of faces (F) = 6

Number of edges (E) = 12

From Euler’s Method,

Number of vertices (V) = E + 2 – F

= 12 + 2 – 6

= 8

So number of vertices for given figure is 8.

Question 2: Find the number of vertices present for a 3D cylinder that is having 2 faces (Top and Bottom are covered) and 0 edges.

Solution:

Given

Number of faces (F) = 2

Number of edges (E) = 0

From Euler’s Method,

Number of vertices (V) =  E + 2 – F

= 0 + 2 – 2

= 0

So number of vertices for given figure is 0.

Vertex of Parabola

In Parabola, the vertex is a point where it actually turns. This is also called a minimum point/maximum point. When the parabola opens down the vertex is called as maximum point else minimum point.

There are two ways of finding the vertex in a parabola based on the given form of the equation.

If the given equation of a parabola is of form ax²+bx+c then vertex of the parabola is given by-

V = (-b/2a, -D/4a)

Where 

D = b² – 4ac

If the given equation of a parabola is of form y = a(x – h)² + k, then the vertex of the parabola is given by-

V = (h , k)

Let’s look into a few examples in finding the vertex of the parabola.

Question 1: Find the vertex of parabola if the equation of parabola is 3x² + x – 2.

Solution:

Given 

Equation of parabola 3x²+x-2

It is of form ax²+bx+c where a=3, b=1, c=-2

So vertex of parabola is V=(-b/2a,-D/4a)

Discriminant can be calculate by formula D=b²-4ac

D=1²-4×3×(-2)

=1-(-24)

=1+24

D=25

Vertex (V)=(-b/2a,-D/4a)

=(-1/2(3),-25/4(3))

V=(-1/6,-25/12)

Hence the vertex of parabola 3x²+x-2 is at (-1/6,-25/12)

Question 2: What is the vertex of parabola if the equation of parabola is x²-4x+3.

Solution:

Given

Equation of parabola is x²-4x+3

It is of form ax²+bx+c where a=1, b=-4, c=3

So vertex of parabola is V=(-b/2a,-D/4a)

Discriminant can be calculate by formula D=b²-4ac

D=(-4)²-4×1×3

=16-12

D=4

Vertex (V)=(-b/2a,-D/4a)

=(-(-4)/2(1),-4/4(1))

=(4/2,-4/4)

V=(2,-1)

Hence the vertex of parabola x2-4x+3 is at (2,-1)

Question 3: Find the vertex of parabola if the equation of parabola is y = 3(x-4)²+2.

Solution:

Given

Equation of parabola is y=3(x-4)²+2

It is of form y=a(x-h)²+k where a=3, h=4, k=2

So vertex of parabola is V=(h,k)

Vertex (V)=(h , k)

=(4,2)

Hence the vertex of parabola 3(x-4)²+2 is at (4,2)

Question 4: what is the vertex of parabola if the equation of parabola is y = 2x²-8x+9.

Solution:

Given

Equation of parabola is y=2x²-8x+9

This can be rewritten into y=2(x-2)²+1

It is of form y=a(x-h)²+k where a=2, h=2, k=1

So vertex of parabola is V=(h,k)

Vertex (V)=(h , k)

=(2,1)

Hence the vertex of parabola 2x²-8x+9 is at (2,1)