Question

Rewrite the quadratic function  in standard form.

Answer 1

• Standard form of a quadratic function :

A quadratic function f (x) = ax² + bx + c can be expressed in the standard form f ( x ) = a(x - h)² + k By completing the square.

• Once in standard form, the vertex is given by (h, k).
• The parabola opens up if a > 0 and opens down if a < 0.

Steps to put quadratic function in standard form :

1. Make sure coefficient of x² is 1.if the leading term is ax² , where a ≠ 1, then factor a out of each x term.
2. Next, take one of the coefficient of x and square it.In other words, [ (1/2).coefficient of x ]² .
3. Add the result of step 2 inside the parenthesis.
4. In order not to change the problem we must subtract(a - result of step 2)outside the parenthesis.
5. Factor the polynomial in parenthesis as a perfect square and simplify any constants.

The quadratic function is f ( x ) = (1/2)x² - 6x + 2.

Step 1 :

The coefficient of x² is 1/2, so, factor 1/2 out of each x term.

f ( x ) = (1/2)[ x² - (2*6)x + (2*2) ]

2f ( x ) = x² - (2*6)x + (2*2)

2f ( x ) = x² - 12x + 4.

Step 2 :

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression

 Here x coefficient = - 12. so, (half the x coefficient)² = (- 12/2)²= 36.

Add 36 to each side

2f ( x ) + 36 = x² - 12x  + 36 + 4.

Step 3:

2f ( x ) + 36 = (x - 6)² + 4.

Step 4:

2f ( x ) = (x - 6)² + 4 - 36

2f ( x ) = (x - 6)² - 32

Step 5:

f ( x ) = (1/2)[ (x - 6)² - 32 ]

f ( x ) = (1/2)(x - 6)² - 16.

This is the standard form of the given quadratic function.