Question

what does the transformation $f(x) \mapsto -f(x)$ do to the graph of $f(x)$?
reflects it across the $y$-axis
shrinks it vertically
reflects it across the $x$-axis
stretches it vertically

Explanation:

For any point $(x, y)$ on the graph of $f(x)$, the transformation $f(x) \mapsto -f(x)$ maps it to $(x, -y)$. This flips the graph over the horizontal x-axis, as the y-coordinate is negated while the x-coordinate stays the same. Reflecting across the y-axis would be $f(x) \mapsto f(-x)$, vertical stretching/shrinking involves multiplying by a positive constant, not -1.

Answer:

reflects it across the x-axis