To convert the vertex form of the equation of a parabola to its standard form, we need to expand the given expression. The vertex form of the equation provided is [tex]\( y = (x-4)^2 + 22 \)[/tex].
Step-by-step, we proceed as follows:
1. Expand the square:
[tex]\[
(x - 4)^2
\][/tex]
By applying the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(x - 4)^2 = x^2 - 2 \cdot 4 \cdot x + 4^2 = x^2 - 8x + 16
\][/tex]
2. Add the constant term:
[tex]\[
y = x^2 - 8x + 16 + 22
\][/tex]
3. Combine like terms:
[tex]\[
y = x^2 - 8x + (16 + 22) = x^2 - 8x + 38
\][/tex]
Thus, the standard form of the equation is:
[tex]\[
y = x^2 - 8x + 38
\][/tex]
Comparing this with the choices provided:
- A. [tex]\( y = x^2 + x + 11 \)[/tex]
- B. [tex]\( y = 4x^2 - 8x + 38 \)[/tex]
- C. [tex]\( y = x^2 - 8x + 38 \)[/tex]
- D. [tex]\( y = x^2 + 8x + 22 \)[/tex]
The correct answer is:
[tex]\[
\boxed{C}
\][/tex]
