To convert the quadratic function \( f(x) = x^2 - 8x + 11 \) into vertex form, we need to rewrite it in the form of \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
Here are the steps to find \( h \) and \( k \):
1. Identify the coefficients \( a \), \( b \), and \( c \) from the standard form \( ax^2 + bx + c \). For \( f(x) = x^2 - 8x + 11 \):
[tex]\[
a = 1, \quad b = -8, \quad c = 11
\][/tex]
2. Calculate \( h \) using the formula \( h = -\frac{b}{2a} \):
[tex]\[
h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\][/tex]
3. Calculate \( k \) using the formula \( k = c - \frac{b^2}{4a} \):
[tex]\[
k = 11 - \frac{(-8)^2}{4 \cdot 1} = 11 - \frac{64}{4} = 11 - 16 = -5
\][/tex]
So, the vertex of the parabola is \((h, k) = (4, -5)\).
Therefore, the quadratic function \( f(x) = x^2 - 8x + 11 \) written in vertex form is:
[tex]\[
f(x) = (x - 4)^2 - 5
\][/tex]
Among the given choices, the correct one is:
[tex]\[
f(x) = (x - 4)^2 - 5
\][/tex]
