Vertex Formula and its Various Examples

Vertex Formula

You will come across the topic of vertex formula when you are going through the topic of conic sections i.e., when you are learning about parabolas, hyperbolas, and ellipses. Students usually find this topic very confusing. However, it is very easy to understand. In the chapter on conic sections, the vertex formula of a parabola is applied when we need to locate the coordinates of a point at which the parabola crosses its axis of symmetry. The coordinates of the vertex of a parabola is often represented as (h,k). We have learned in the chapter on quadratic equation that its standard form is ax.x + bx + c. Likewise, the standard form of a parabola is the same i.e., ax.x + bx + c. Now, we have the vertex form of a parabola as a(x-h) (x-h) + k. In this article, we will discuss in detail the methods of determining the vertex of a parabola and solve some examples related to it.

Various Methods to Determine the Vertex of a Parabola 

There are two different ways through which we can determine the vertex of a parabola which is discussed as follows:

  • We can determine the vertex (h,k) by taking (-b/ 2a, -D/ 4a) where D represents the discriminant and is equal to b.b – 4ac.
  • We can determine the vertex (h,k) by taking h = -b/2a and then evaluate y at h to calculate k.

Some Solved Examples of Vertex Formula

Example 1: Find out the vertex of the given parabola, y = 2x.x – 5x + 1.

Answer: We need to find out the vertex of the given equation.

By comparing the equation y = 2x.x – 5x + 1, we get a = 2, b = -5, c = 1.

Now, let us find out the value of discriminant.

D = b.b – 4ac = (-5) (-5) – 4 * 2 * 1 = 25 – 8 = 17

Now, using the vertex formula, we get: 

(h, k) = (-b/ 2a, -D/ 4a) = { -(-5) / 2 * 2, -17 / 4 * 2} = (5/4, -17/8)

Thus, the vertex of the given parabola is (5/4, -17/8).

 

Example 2: Find out the vertex of the given parabola, y = 2x.x – 6x + 1.

Answer: We need to find out the vertex of the given equation.

By comparing the equation y = 2x.x – 6x +1, we get a = 2, b = -6, c = 1.

Now, let us find out the value of discriminant.

D = b.b – 4ac = (-6) (-6) – 4 * 2 * 1 = 36 – 8 = 28

Now, using the vertex formula, we get: 

(h, k) = (-b/ 2a, -D/ 4a) = { -(-6) / 2 * 2, -28 / 4 * 2} = (6/4, -28/8)

Thus, the vertex of the given parabola is (6/4, -28/8).

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