To determine the resulting function when the function [tex]\( f(x) = x^2 - 7 \)[/tex] is vertically stretched by a factor of 2 and then translated downward by 5 units, let's go through each transformation step by step.
1. Vertically stretch by a factor of 2:
When a function [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 2, every y-value is multiplied by 2. So, the new function after vertical stretching is:
[tex]\[
y = 2 \cdot (x^2 - 7)
\][/tex]
Simplifying this expression:
[tex]\[
y = 2x^2 - 14
\][/tex]
2. Translate downward by 5 units:
When a function is translated downward by 5 units, we subtract 5 from every y-value. So, the new function after this translation is:
[tex]\[
y = 2x^2 - 14 - 5
\][/tex]
Simplifying this expression:
[tex]\[
y = 2x^2 - 19
\][/tex]
Having performed both transformations, the resulting function is:
[tex]\[
y = 2x^2 - 19
\][/tex]
Now, let's compare this result with the given options:
A) [tex]\( y = 2x^2 - 2 \)[/tex]
B) [tex]\( y = 2x^2 - 12 \)[/tex]
C) [tex]\( y = 2x^2 - 5 \)[/tex]
D) [tex]\( y = x^2 - 12 \)[/tex]
None of the provided options match the result [tex]\( y = 2x^2 - 19 \)[/tex]. Therefore, the answer is that none of the given options are correct.
Following through the entire process step-by-step, none of the provided options are the correct transformed function. Thus, the correct answer among the given choices is:
[tex]\[
\boxed{-1}
\][/tex]
