Answered

The image shows a geometric representation of the function [tex]f(x)=x^2-2x-6[/tex] written in standard form.

What is this function written in vertex form?

A. [tex]f(x) = (x-1)^2 - 7[/tex]
B. [tex]f(x) = (x+1)^2 - 7[/tex]
C. [tex]f(x) = (x-1)^2 - 5[/tex]
D. [tex]f(x) = (x+1)^2 - 5[/tex]



Answer :

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.