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VOOZH | about |
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VOOZH | about |
In mathematics and geometry, a vertex is a fundamental concept that refers to a point where two or more curves, edges, or lines meet. The term is used in various contexts and has specific meanings depending on the geometric or algebraic structure being considered.
In a polygon (such as a triangle, quadrilateral, etc.), a vertex is a point where two sides of the polygon meet. For an angle, a vertex is the point where the two rays or line segments that form the angle meet.
A vertex is denoted by capital letters like A, B, E etc. In geometry, there are many shapes like cubes, squares, triangles etc. For these figures, there is more than one vertex. So the plural form of a vertex is called vertices.
Let's look at a few figures added below:
👁 Vertex DefinitionIn the above figure for ∠A, 'A' is the vertices, in triangle ABC A, B and C are vertices and in pentagon ABCDE, A, B, C, D, and E are vertices.
There is an Euler formula to calculate how many vertices are present in 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,
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.
👁 Vertex of ParabolaThere 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,
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 at the few questions on finding the number of vertices for the given figures and also vertex of parabola.
Example 1: Find the number of vertices present for a figure (cube) with 6 faces and 12 edges.
👁 Example 1 on VertexSolution:
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.
Example 2: Find the number of vertices present for a 3D cylinder that is having 2 faces (Top and Bottom are covered) and 0 edges.
👁 Example 2 on VertexSolution:
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.
Example 3: 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)
Example 4: 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)
Example 5: 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)
Example 6: 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)
