To convert the function [tex]\( f(x) = x^2 - 2x - 6 \)[/tex] from standard form to vertex form, we need to complete the square. Here's a step-by-step solution:
1. Start with the standard form of the quadratic function:
[tex]\[
f(x) = x^2 - 2x - 6
\][/tex]
2. Isolate the quadratic and linear terms:
[tex]\[
f(x) = (x^2 - 2x) - 6
\][/tex]
3. Complete the square:
- To complete the square, we need to add and subtract the same value inside the parenthesis. This value is the square of half the coefficient of [tex]\( x \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\(-2\)[/tex], so half of that is [tex]\(-1\)[/tex], and its square is [tex]\((-1)^2 = 1\)[/tex].
Adding and subtracting this value inside the parenthesis:
[tex]\[
f(x) = (x^2 - 2x + 1) - 1 - 6
\][/tex]
4. Rewrite the quadratic expression as a square of a binomial:
- The expression [tex]\( x^2 - 2x + 1 \)[/tex] can be written as [tex]\( (x - 1)^2 \)[/tex].
So, we have:
[tex]\[
f(x) = (x - 1)^2 - 1 - 6
\][/tex]
5. Combine the constants:
[tex]\[
f(x) = (x - 1)^2 - 7
\][/tex]
Therefore, the function [tex]\( f(x) = x^2 - 2x - 6 \)[/tex] written in vertex form is [tex]\( f(x) = (x - 1)^2 - 7 \)[/tex].
Thus, the correct answer is:
[tex]\[
f(x) = (x - 1)^2 - 7
\][/tex]
