## STANDARD FORM TO VERTEX FORM BY COMPLETING THE SQUARE

Standard Form to Vertex Form by Completing the Square :

In this section, you will learn how to convert the standard form of a quadratic function into vertex form by completing the square.

The standard form a quadratic function is

y  =  ax2 + bx + c

The vertex form a quadratic function is

y  =  a(x - h)2 + k

Here, the vertex is (h, k).

## Standard Form to Vertex Form by Completing the Square - Steps

Step 1 :

In the given quadratic function y  =  ax2 + bx + c, factor "a" from the first two terms of the quadratic expression on the right side.

Then,

y  =  ax2 + bx + c

y  =  a(x2 + bx/a) + c

Note :

If the coefficient of x2 is 1 (a = 1), the above process is not required.

Step 2 :

In the result of step 1, write the "x" term as a multiple of 2.

Examples :

6x should be written as 2(3)(x).

5x should be written as 2(x)(5/2).

Then, the result of this step will be :

y  =  a[x2 + 2(x)(b/a)] + c

Step 3 :

Now add and subtract (b/a)2 inside the parentheses to complete the square on the left side.

Then,

y  =  a[x2 + 2(x)(b/a) + (b/a)2 - (b/a)2] + c

Step 4 :

In the result of step 3, if we use the algebraic identity

(a + b)2  =  a2 + 2ab + b2

inside the parentheses, we get

y  =  a[(x + b/a)2 - (b/a)2] + c

Simplify.

y  =  a[(x + b/a)2 - b2/a2] + c

y  =  a(x + b/a)2 - b2/a + c

y  =  a(x + b/a)2 - b2/a + ac/a

y  =  a(x + b/a)2 + (ac - b2)/a

The above quadratic is in the form of

y  =  a(x - h)+ k

Here, the vertex is (h, k).

## Standard Form to Vertex Form by Completing the Square - Examples

Example 1 :

Write the following quadratic function in vertex form and sketch the parabola.

y  =  x2 - 4x + 3

Solution :

Step 1 :

In the quadratic function given, the coefficient of x2 is 1. So, we can skip step 1.

Step 2 :

In the quadratic function y  =  x2 - 4x + 3, write the "x" term as a multiple of 2.

Then,

y  =  x2 - 2(x)(2) + 3

Step 3 :

Now add and subtract 22 on the right side to complete the square.

Then,

y  =  x2 - 2(x)(2) + 22 - 22 + 3

y  =  x2 - 2(x)(2) + 22 - 4 + 3

y  =  x2 - 2(x)(2) + 22 - 1

Step 4 :

In the result of step 3, if we use the algebraic identity

(a - b)2  =  a2 - 2ab + b2

on the right side, we get

y  =  (x - 2)2 - 1

The quadratic function above is in vertex form.

Comparing

y  =  (x - 2)2 - 1

and

y  =  a(x - h)2 + k,

the vertex is

(h, k)  =  (2, -1)

and

a  =  1

Graph of the Parabola :

The vertex of the parabola is (2, -1). Because the sign of "a" is positive the parabola opens upward. Example 2 :

Write the following quadratic function in vertex form and sketch the parabola.

y  =  2x2 - 8x + 9

Solution :

Step 1 :

In the quadratic function given, the coefficient of x2 is 2. So, factor "2" from the first two terms of the quadratic expression on the right side.

y  =  2(x2 - 4x) + 9

Step 2 :

In the quadratic function y  =  2(x2 - 4x) + 9write the "x" term as a multiple of 2.

Then,

y  =  2[x2 - 2(x)(2)] + 9

Step 3 :

Now add and subtract 22 inside the parentheses to complete the square.

Then,

y  =  2[x2 - 2(x)(2)+ 22 - 22] + 9

y  =  2[x2 - 2(x)(2)+ 22 - 4] + 9

Step 4 :

In the result of step 3, if we use the algebraic identity

(a - b)2  =  a2 - 2ab + b2

inside the parentheses, we get

y  =  2[(x - 2)2 - 4] + 9

y  =  2(x - 2)2 - 8 + 9

y  =  2(x - 2)2 + 1

The quadratic function above is in vertex form.

Comparing

y  =  2(x - 2)2 + 1

and

y  =  a(x - h)2 + k,

the vertex is

(h, k)  =  (2, 1)

and

a  =  2

Graph of the Parabola :

The vertex of the parabola is (2, 1). Because the sign of "a" is positive the parabola opens upward.  After having gone through the stuff given above, we hope that the students would have understood how to write a quadratic function in vertex form when it is given in standard  form.

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