Understanding a Function Transformation
The image contains a question regarding the transformation of the function \( f(x) = x^2 \) when modified to \( g(x) = 4 f(x) - 4x^2 \).
Given the function \( g(x) = 4 f(x) - 4x^2 \), we can substitute \( f(x) \) with \( x^2 \) to rewrite \( g(x) \) as:
\( g(x) = 4(x^2) - 4x^2 \)
This simplifies to:
\( g(x) = 4x^2 - 4x^2 \)
\( g(x) = 0 \)
The graph of \( g(x) \) will thus be a horizontal line at \( y = 0 \), which is not one of the given options. However, it is clear that the original transformation intended before simplification is equivalent to multiplying the function \( f(x) \) by 4, which would stretch the graph vertically by a factor of 4 and then subtracting \( 4x^2 \) would do nothing since it cancels out the stretching. It seems there might be a mistake in the question since the transformation does not lead to a vertical stretch, shift, or compression but rather to the graph being a horizontal line.
However, if we consider the given options and interpret the transformation as it might have been intended, to be \( g(x) = 4 \cdot (f(x) - x^2) \), then the correct answer would be:
A. The graph of \( g(x) \) is the graph of \( f(x) \) stretched vertically by a factor of 4.
This interpretation involves taking the original function \( f(x) = x^2 \), subtracting \( x^2 \) from it (which, again, simplifies to zero), and multiplying the result by 4. But as previously noted, there seems to be an error in the question since the transformation provided does not result in a visual change to the graph of \( f(x) \) as given by the options.
