To convert standard form to vertex form, we may follow the different ways.
From the standard form of the equation, y = ax^{2} + bx + c
(i) Take the coefficient of x^{2}, from all the terms if there is.
(ii) Write the coefficient of x as a multiple of 2.
(iii) Get any one of the algebraic identities (a+b)^{2} or (a-b)^{2}
Example :
Convert the following quadratics from standard form to vertex form
y = 2x^{2} – 4x + 8
Solution :
y = 2x^{2} – 4x + 8
Factor 2.
y = 2(x^{2} – 2x + 4)
Write the coefficient of x as a multiple of 2.
y = 2(x^{2} – 2⋅x⋅1 + 1^{2 }- 1^{2} + 4)
Here x^{2} – 2⋅x⋅1 + 1^{2} matches with a^{2} – 2⋅a⋅b + b^{2} = (a - b)^{2}
y = 2[(x - 1)^{2} - 1 + 4]
y = 2[(x - 1)^{2} + 3]
Distributing 2, we get
y = 2(x - 1)^{2} + 6
From the standard form of the equation, y = ax^{2} + bx + c
(i) Take the coefficient of x^{2}, from all the terms if there is.
(ii) Take half of the coefficient of x and write it as (x - a)^{2} or (x + a)^{2. }Here a is half the coefficient of x.
Example :
Convert y = x^{2} - 4x + 3 into factored form.
Solution :
y = x^{2} - 4x + 3
Coefficient of x^{2} is 1, so dont have to factor anything.
Half of the coefficient of x is 2.
y = (x - 2)^{2} - 2^{2} + 3
y = (x - 2)^{2} - 4 + 3
y = (x - 2)^{2} - 1
Complete the square to convert the standard form quadratic function into vertex form. Then
find the vertex.
Example 1 :
f(x) = x^{2} + 4x - 14
Solution :
Coefficient of x^{2} is 1. So, don't have to factorize.
Write the coefficient of x as multiple of 2.
f(x) = x^{2} + 2 ⋅ x ⋅ 2 + 2^{2} - 2^{2 } - 14
f(x) = (x+2)^{2} - 4 - 14
f(x) = (x+2)^{2} - 18
Example 2 :
f(x) = 2x^{2} + 9x
Solution :
Coefficient of x^{2} is 2.
Write the coefficient of x as multiple of 2.
Example 3 :
f(x) = 5x^{2} - 4x + 1
Solution :
Coefficient of x^{2} is 5. So, we have to factorize 5.
Example 4 :
f(x) = x^{2} - 16x + 70
Solution :
Coefficient of x^{2} is 1. So, don't have to factorize.
Write the coefficient of x as multiple of 2.
f(x) = x^{2} - 2 ⋅ x ⋅ 8 + 8^{2} - 8^{2} + 70
f(x) = (x - 8)^{2} - 64 + 70
f(x) = (x - 8)^{2} + 6
Example 5 :
f(x) = -3x^{2} + 48x - 187
Solution :
Coefficient of x^{2} is -3. So, factorize -3.
f(x) = -3[x^{2} - 16x] - 187
Write the coefficient of x as multiple of 2.
f(x) = -3[x^{2} - 2 ⋅ x ⋅ 8 + 8^{2}- 8^{2}] - 187
f(x) = -3[(x - 8)^{2}- 64] - 187
f(x) = -3(x - 8)^{2}+ 192 - 187
f(x) = -3(x - 8)^{2}+ 5
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