To find the standard form of the given function [tex]\( f(x) = 7(x - 1)^2 + 3 \)[/tex], we will convert it from the vertex form to the standard quadratic form [tex]\( ax^2 + bx + c \)[/tex].
1. Expand the squared term:
[tex]\[
(x - 1)^2 = x^2 - 2x + 1
\][/tex]
2. Distribute the 7 across the expanded quadratic:
[tex]\[
7(x - 1)^2 = 7(x^2 - 2x + 1)
\][/tex]
[tex]\[
= 7x^2 - 14x + 7
\][/tex]
3. Add the constant term 3 to complete the equation:
[tex]\[
f(x) = 7x^2 - 14x + 7 + 3
\][/tex]
[tex]\[
f(x) = 7x^2 - 14x + 10
\][/tex]
Thus, the function [tex]\( f(x) = 7(x - 1)^2 + 3 \)[/tex] in standard form is [tex]\( f(x) = 7x^2 - 14x + 10 \)[/tex].
Now, we compare this with the given choices:
A. [tex]\( f(x) = 7x^2 - 14x - 50 \)[/tex]
B. [tex]\( f(x) = -7x^2 - 14x + 10 \)[/tex]
C. [tex]\( f(x) = -7x^2 - 14x - 10 \)[/tex]
D. [tex]\( f(x) = 7x^2 - 14x + 10 \)[/tex]
Matching our result with the choices, the correct answer is:
D. [tex]\( f(x) = 7x^2 - 14x + 10 \)[/tex]
