To rewrite the quadratic function [tex]\( f(x) = x^2 - 8x + 44 \)[/tex] from standard form to vertex form, we will complete the square. This involves rewriting the quadratic expression in the form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h,k)\)[/tex] is the vertex of the parabola.
Here are the steps to complete the square:
1. Start with the given quadratic function in standard form:
[tex]\[
f(x) = x^2 - 8x + 44
\][/tex]
2. To complete the square, focus on the [tex]\( x^2 \)[/tex] and [tex]\( -8x \)[/tex] terms. Take the coefficient of the [tex]\( x \)[/tex] term, which is [tex]\(-8\)[/tex], divide it by 2, and then square it:
[tex]\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\][/tex]
3. Add and subtract this square (16) inside the function:
[tex]\[
f(x) = x^2 - 8x + 16 - 16 + 44
\][/tex]
4. Rewrite the quadratic expression by grouping the perfect square trinomial and combining the constants:
[tex]\[
f(x) = (x^2 - 8x + 16) + (44 - 16)
\][/tex]
[tex]\[
f(x) = (x - 4)^2 + 28
\][/tex]
5. Thus, the vertex form of the quadratic function is:
[tex]\[
f(x) = (x - 4)^2 + 28
\][/tex]
So, the quadratic function [tex]\( f(x) = x^2 - 8x + 44 \)[/tex] rewritten in vertex form is [tex]\( f(x) = (x - 4)^2 + 28 \)[/tex].
