Question

which function in vertex form is equivalent to ( f(x) = 4 + x^2 - 2x )?
( f(x) = (x - 1)^2 + 3 )
( f(x) = (x - 1)^2 + 5 )
( f(x) = (x + 1)^2 + 3 )
( f(x) = (x + 1)^2 + 5 )

Explanation:

Step1: Recall Vertex Form Formula

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. We need to complete the square for \( f(x) = 4 + x^2 - 2x \). First, rearrange the terms: \( f(x) = x^2 - 2x + 4 \).

Step2: Complete the Square

For the quadratic \( x^2 - 2x \), take half of the coefficient of \( x \) (which is \(-2\)), so \(\frac{-2}{2} = -1\), square it: \((-1)^2 = 1\). Add and subtract this inside the function:

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Answer:

\( f(x) = (x - 1)^2 + 3 \) (corresponding to the first option: \( f(x)=(x - 1)^2 + 3 \))