What is [tex]$f(x)=x^2-8x+11$[/tex] written in vertex form?

A. [tex]$f(x)=(x-4)^2-5$[/tex]
B. [tex]$f(x)=(x-4)^2+5$[/tex]
C. [tex]$f(x)=(x+4)^2-27$[/tex]
D. [tex]$f(x)=(x+4)^2+27$[/tex]



Answer :

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]