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What Is The Vertex Form And How To Calculate It?

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In algebra, a vertex form calculator is widely used to find the vertex of two points. The point at which two lines meet to form an angle is said to be the vertex. The vertex of the parabola is usually used on a wider scale.

The vertex form can be maximum or minimum depending upon the positive or negative values of the vertex (h, k). In this article, we will learn the definition, formula, and examples of the vertex form.

What is the vertex form?

In geometry, a point where two edges or lines meet to form an angle is said to be the vertex. For example, a pentagon has 5 cornersand each corner is known as the vertex. In other words, the intersecting point of x & y coordinates of the parabola is said to be the vertex.

In a parabola graph, the vertex is the extremal point. The vertex can be maximum or minimum. The points of the vertex form are h & k. It is calculated easily just by putting the values of h & k. The vertex form can also be derived from the equation of a parabola.

As you know that the equation of the standard form involves the coefficients a, b, & c. So, you have to find the points of vertex (h, k) with the help of coefficients of standard form and put them in the equation of vertex form.

Formula of the vertex form

The equation or formula used to find the vertex form is given below.

y = a * (x – h)2 + k

where x & y are the coordinate points, h & k are the vertex point, and a is the constant coefficient of the equation.

The vertex form can also be determined by using the equation of the standard form.

The equation of standard form is:

ax2 + bx + c = 0

where, a, b, & c are the coefficients of the equation. To find the vertex form, find the h & k points by using a, b, & c. The formulas used to find h & k are given below.

h = -b / 2a

k = c – [b2] / [4a]

How to calculate the vertex form?

The vertex form can be calculated easily by using the formulas of vertex form calculator and the standard form. Below are solved examples of the vertex form.

Example 1: For vertex form

Find the vertex form if h = 4, k = 6, & a = 4.

Solution

Step 1:Write the given terms.

h = 4

k = 6

a = 4

Step 2:Take the general equation of the vertex form.

y = a * (x – h)2 + k

Step 3:Place the given terms in the equation of the vertex form.

y = a * (x – h)2 + k

y = 4 * (x – 4)2 + 6

Hence, this is the required vertex form.

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Example 2:

Find the vertex form if h = 5.7, k = 8.9, & a = 7.

Solution

Step 1:Write the given terms.

h = 5.7

k = 8.9

a = 7

Step 2:Take the general equation of the vertex form.

y = a * (x – h)2 + k

Step 3:Place the given terms in the equation of the vertex form.

y = a * (x – h)2 + k

y = 7 * (x – 5.7)2 + 8.9

Hence, this is the required vertex form.

Example 3: For standard form

Find the vertex form if the equation of the standard form is 6x2 + 8x + 9 = 0.

Solution

Step 1:Write the given equation and the coefficients of the equation.

y = 6x2 + 8x + 9

a = 6

b = 8

c = 9

Step 2:Now take the general formulas to find the vertex points h & k.

h = -b / 2a

k = [c – b2] / [4a]

Step 3:Now place the values of a & b in the formula of h to calculate it.

h = -b / 2a

h = -8 / 2(6)

h = -8 / 12

h = -2 / 3

h = – 0.67

Step 4:Substitute the values of a, b, &c in the formula of k to calculate it.

k = c – [b2] / [4a]

k = 9 –[82] / [4(6)]

k = 9 –[64] / [4(6)]

k = 9 –[64] / [24]

k = 9 – 8 / 3

k = 9 – 2.67

k = 6.33

Step 5:Take the general equation of the vertex form.

y = a * (x – h)2 + k

Step 6:Place the calculated points in the equation of the vertex form.

y = 6 * (x – (-0.67))2 + 6.33

y = 6 * (x + 0.67)2+ 6.33

You can also use a vertex form calculator to solve the above problem. Follow the below steps to calculate the vertex form by using this tool.

Step 1: Select the method i.e., vertex form or standard form.

Step 2: Enter the required values into the input boxes.

Step 3: Press the calculate button.

Example 4

Find the vertex form if the equation of the standard form is 8x2 – 12x – 14 = 0.

Solution

Step 1:Write the given equation and the coefficients of the equation.

y = 8x2 – 12x – 14

a = 8

b = -12

c = -14

Step 2:Now take the general formulas to find the vertex points h & k.

h = -b / 2a

k = c – [b2] / [4a]

Step 3:Now place the values of a & b in the formula of h to calculate it.

h = -b / 2a

h = -(-12) / 2(8)

h = 12 / 16

h = 6 / 8 = 3 / 4

h = 0.75

Step 4:Substitute the values of a, b, &c in the formula of k to calculate it.

k = c – [b2] / [4a]

k = -14 – [-122] / [4(8)]

k = -14 – [144] / [4(8)]

k = -14 – [144] / [32]

k = -14 – 18/ 4

k = -14 – 9 / 2

k = -14 – 4.5

k = -18.5

Step 5:Take the general equation of the vertex form.

y = a * (x – h)2 + k

Step 6:Place the calculated points in the equation of the vertex form.

y = 6 * (x – 0.75)2 + (-18.5)

y = 6 * (x – 0.75)2– 18.5

Summary

The vertex form is widely used in algebra and geometry. Now you can grab all the basics of the vertex form from this post. Once you grab all the basics of this topic, you can easily solve any problem of this topic.