To determine the correct function resulting from vertically stretching [tex]\( f(x) = x^2 - 7 \)[/tex] by a factor of 2 and then translating it downward 5 units, follow these detailed steps:
1. Original Function: Start with the given function [tex]\( f(x) = x^2 - 7 \)[/tex].
2. Vertically Stretch: Applying a vertical stretch by a factor of 2 means you multiply the entire function by 2.
[tex]\[
g(x) = 2 \cdot (x^2 - 7)
\][/tex]
Simplifying [tex]\( g(x) \)[/tex], we get:
[tex]\[
g(x) = 2x^2 - 14
\][/tex]
3. Translate Downward: Translating the function downward by 5 units means subtracting 5 from the entire function.
[tex]\[
h(x) = g(x) - 5 = (2x^2 - 14) - 5
\][/tex]
Simplifying [tex]\( h(x) \)[/tex], we get:
[tex]\[
h(x) = 2x^2 - 14 - 5
\][/tex]
[tex]\[
h(x) = 2x^2 - 19
\][/tex]
4. Conclusion: The final function after these transformations is [tex]\( h(x) = 2x^2 - 19 \)[/tex].
Now, 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 these provided options match the final function [tex]\( h(x) = 2x^2 - 19 \)[/tex].
Therefore, the correct conclusion is that none of the provided options accurately represent the resultant function after the described transformations.
