To determine which function in vertex form is equivalent to [tex]\( f(x) = x^2 + 8 - 16x \)[/tex], we need to rewrite the given quadratic function in its vertex form. Vertex form of a quadratic function is given by:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
Where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
First, let's rewrite the given function in a standard form:
[tex]\[ f(x) = x^2 - 16x + 8 \][/tex]
Next, we complete the square to transform it into vertex form.
1. Start with the original function:
[tex]\[ f(x) = x^2 - 16x + 8 \][/tex]
2. Group the [tex]\(x\)[/tex] terms together:
[tex]\[ f(x) = (x^2 - 16x) + 8 \][/tex]
3. To complete the square, add and subtract the square of half the coefficient of [tex]\(x\)[/tex] inside the parentheses. The coefficient of [tex]\(x\)[/tex] is [tex]\(-16\)[/tex], and half of [tex]\(-16\)[/tex] is [tex]\(-8\)[/tex], so we square it to get [tex]\(64\)[/tex]:
[tex]\[ f(x) = (x^2 - 16x + 64 - 64) + 8 \][/tex]
[tex]\[ f(x) = (x^2 - 16x + 64) - 64 + 8 \][/tex]
4. Now, the expression inside the parentheses is a perfect square trinomial:
[tex]\[ f(x) = (x - 8)^2 - 64 + 8 \][/tex]
[tex]\[ f(x) = (x - 8)^2 - 56 \][/tex]
Thus, the vertex form of the given quadratic function [tex]\( f(x) = x^2 - 16x + 8 \)[/tex] is:
[tex]\[ f(x) = (x - 8)^2 - 56 \][/tex]
Let's compare this with the given options:
1. [tex]\( f(x) = (x - 8)^2 - 56 \)[/tex] [tex]\(\Rightarrow\)[/tex] This matches our vertex form.
2. [tex]\( f(x) = (x - 4)^2 + 0 \)[/tex]
3. [tex]\( f(x) = (x + 8)^2 - 72 \)[/tex]
4. [tex]\( f(x) = (x + 4)^2 - 32 \)[/tex]
Therefore, the function in vertex form that is equivalent to [tex]\( f(x) = x^2 + 8 - 16x \)[/tex] is:
[tex]\[ f(x) = (x - 8)^2 - 56 \][/tex]
