To convert the given quadratic function from standard form \( f(x) = x^2 - 2x - 6 \) to vertex form, we'll use the method of completing the square. Here’s a detailed, step-by-step solution:
1. Rewrite the function as \( ax^2 + bx + c \):
[tex]\[
f(x) = x^2 - 2x - 6
\][/tex]
Here, \(a = 1\), \(b = -2\), and \(c = -6\).
2. Isolate the quadratic and linear terms:
[tex]\[
f(x) = (x^2 - 2x) - 6
\][/tex]
3. Complete the square:
To complete the square, first find the value that completes the square for the quadratic expression inside the parentheses. The term needed to complete the square is \(\left(\frac{b}{2}\right)^2\).
[tex]\[
\left(\frac{-2}{2}\right)^2 = 1
\][/tex]
Add and subtract this value inside the parentheses:
[tex]\[
f(x) = (x^2 - 2x + 1 - 1) - 6
\][/tex]
Combine the complete square and the remaining terms:
[tex]\[
f(x) = [(x - 1)^2 - 1] - 6
\][/tex]
4. Simplify the expression:
Combine the constants outside the square:
[tex]\[
f(x) = (x - 1)^2 - 1 - 6
\][/tex]
[tex]\[
f(x) = (x - 1)^2 - 7
\][/tex]
Therefore, the function in vertex form is:
[tex]\[
f(x) = (x - 1)^2 - 7
\][/tex]
So, the correct choice is:
- \( f(x) = (x - 1)^2 - 7 \)
This corresponds to the given result that validates our solution.
