To find the equivalent function in vertex form for [tex]\( f(x) = 4 + x^2 - 2x \)[/tex], we will start by completing the square. Here are the detailed steps to transform the given quadratic function into its vertex form:
1. Rearrange the function:
[tex]\[
f(x) = x^2 - 2x + 4
\][/tex]
2. Identify the coefficient of [tex]\(x\)[/tex]:
The coefficient of [tex]\(x\)[/tex] in the quadratic term [tex]\( x^2 - 2x \)[/tex] is [tex]\(-2\)[/tex].
3. Complete the square:
To complete the square, we need to add and subtract the square of half the coefficient of [tex]\(x\)[/tex]. Here, half the coefficient of [tex]\(-2\)[/tex] is [tex]\(-1\)[/tex], and [tex]\((-1)^2 = 1\)[/tex].
4. Rewrite the function:
[tex]\[
f(x) = x^2 - 2x + 4 = x^2 - 2x + 1 - 1 + 4
\][/tex]
5. Group the perfect square trinomial:
[tex]\[
f(x) = (x^2 - 2x + 1) + (-1 + 4)
\][/tex]
[tex]\[
f(x) = (x - 1)^2 + 3
\][/tex]
Thus, the vertex form of the function [tex]\( f(x) = 4 + x^2 - 2x \)[/tex] is:
[tex]\[
f(x) = (x - 1)^2 + 3
\][/tex]
So, the correct function in vertex form is:
[tex]\[
f(x) = (x-1)^2 + 3
\][/tex]
From the given options, the correct answer is:
[tex]\[
\boxed{f(x) = (x-1)^2 + 3}
\][/tex]
