# CONVERTIG FROM VERTEX FORM TO STANDARD FORM

To convert standard form to vertex form, we may follow the different ways. We should aware of the following algebraic identities.

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

Convert the following quadratics from vertex form to standard form.

Problem 1 :

y = -(x – 1)2 – 1

Solution :

y = -(x – 1)2 – 1

Using the algebraic identity, we can expand this and then distribute the negative sign.

(x – 1)2 = x2 - 2x(1) + 12

(x – 1)2 = x2 - 2x + 1

y = -(x2 - 2x + 1) – 1

Distributing the negative, we get

y = -x2 + 2x - 1 – 1

y = -x2 + 2x - 2

Problem 2 :

y = 2(x – 2)2 – 3

Solution :

y = 2(x – 2)2 – 3

Using the algebraic identity, we can expand it and multiply by 2.

(x – 2)2 = x2 - 2x(2) + 22

(x – 2)2 = x2 - 4x + 4

y = 2(x2 - 4x + 4) – 3

Distributing 2, we get

y = 2x2 - 8x + 8 – 3

Combining the like terms, we get

y = 2x2 - 8x + 5

Problem 3 :

y = (x + 4)2 + 4

Solution :

y = (x + 4)2 + 4

Using the algebraic identity, we can expand it and multiply by 2.

(x + 4)2 = x2 - 2x(4) + 42

(x – 1)2 = x2 - 8x + 16

y = (x2 - 8x + 16) + 4

Combining the like terms, we get

y = x2 - 8x + 20

Problem 4 :

y = (1/2)(x - 2)2 - 4

Solution :

y = (1/2)(x - 2)2 - 4

Using the algebraic identity, we can expand it and multiply by 2.

(x - 2)2 = x2 - 2x(2) + 22

(x – 2)2 = x2 - 4x + 4

y = (1/2)(x2 - 4x + 4) - 4

Distributing 1/2, we get

y = (1/2)x2 - 2x + 2 - 4

y = (1/2)x2 - 2x - 2

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