Vertex Formula

In geometry vertex definition states that it is a point where two or more curves, lines, or edges meet.

The below diagram shows the vertex form in 2D and 3D shapes.

Vertex

A vertex is a corner point formed by the junction of edges, faces, or facets of a polygon, polyhedron, or any higher-dimensional polytope.

If the internal angle of the polygon, which is the angle produced by the two edges at the vertex with the polygon inside the angle, is less than radians, the vertex is considered convex; otherwise, it is called concave or reflex. A vertex of a polyhedron or polytope is convex if its intersection with a suitably tiny sphere centred at the vertex math is convex, and is concave otherwise.

Vertex Examples:

• The polygon shapes are formed by joining the sides at the point at each corner called the vertex. A triangle has 3 corners and each corner is called a vertex. 

• When two rays cross each other at a point with a certain angle then the point where the two rays meet is called a vertex.

• In a graph, the 2 axis x and y starts at the point called origin which is also a vertex.

How to Find Vertex Formula?

Here we will learn the step by step derivation of the Vertex formula 

To find the vertex math formula of a parabola, consider the standard form of a parabola equation 

y = ax2 + bx + c …………………(1)

Now converting the standard form of the parabola to the vertex form by completing the square we get,

y = a(x - h)2 + k……………..(2)

Vertex

Now subtracting by ‘c’ on both sides of equation (1) we get

y - c = ax2 + bx + c - c

y -c = ax2 + bx…………..(3)

Now in the RHS of equation (3) take ‘a’ as a common factor,

$y-c=a[ x^2+bx/a]$………………(4)

Now add and subtract the RHS of equation (4) by $b^{2}/4a^{2}$

$\\y-c=a[ x^2+bx/a +b^{2}/4a^{2} -b^{2}/4a^{2}]$…………..(5)

Now using the algebraic identity (a + b)2 = a2 + b2.+2ab we can simplify the equation (5) as follows:

$\\y-c=a[\lgroup x+b/2a\rgroup^2-b^{2}/4a^{2}]$……………(6)

Now adding ‘c’ on both sides of the equation (6) and simplifying we get

$\\y-c+c=a\lgroup x+b/2a\rgroup^2-b^{2}a/4a^{2}+c$

$\\y=a\lgroup x+b/2a\rgroup^2-\left(\frac{b^2-4ac}{4a}\right)$……………….(7)

Now comparing equation (7) with equation (2) we get

h=-b/2a

$k=-\left(\frac{b^2-4ac}{4a}\right)$

Therefore the vertex formula according to vertex definition will be

$\lgroup h,k\rgroup=\lgroup -b/2a ,-\left(\frac{b^2-4ac}{4a}\right)\rgroup$

Simplifying further we get the vertex formula as 

$\lgroup h,k\rgroup=\lgroup -b/2a ,c-b^{2}/4a\rgroup$

Here h is the x - coordinate of the vertex and k is the y - coordinate of the vertex.

Problems on Vertex Formula

1. Find the vertex of the parabola y = 4x2 + 16x - 20.

Ans: Comparing the given equation of the parabola y = 4x2 + 16x  - 20 with the standard equation of parabola y = ax2 + bx + c we get

a = 4, b = 16 and c = -20

The vertex formula of the parabola is given as 

$\lgroup h,k\rgroup=\lgroup -b/2a ,c-b^{2}/4a\rgroup$

x - coordinate of the vertex is  

h = -b/2a

Substituting the values we get

h=-16/8

h = -2

y - coordinate of the vertex is 

$k=c-b^{2}/4a$

Substituting the values we get

$k=-20-256/16$

k=-20-16 

k=-36

Therefore the vertex of the parabola y = 4x2 + 16x - 20 is (h, k) = (-2, -36).

Conclusion

When two line segments intersect in geometry, the location where the two lines meet is known as a vertex. This is true whether the lines intersect or meet at a corner. As a result, angles have vertices as well. An angle is a measurement of the connection between two line segments, known as rays, that intersect at a certain place. The vertex of a parabola is the exact point at which it turns. It's also known as the minimum point. The maximum point is the vertex of a parabola when it expands down. The axis of symmetry is where the parabola vertex is located.