1. The problem asks to graph the functions $-f(x)$, $f(-x)$, and $-f(-x)$ based on a given function $f(x)$. 2. These transformations relate to reflections of the graph of $f(x)$: - $-f(x)$ reflects the graph of $f(x)$ across the x-axis. - $f(-x)$ reflects the graph of $f(x)$ across the y-axis. - $-f(-x)$ reflects the graph of $f(x)$ across both axes (x-axis and y-axis). 3. To understand these, recall: - Reflecting across the x-axis changes $y$ to $-y$. - Reflecting across the y-axis changes $x$ to $-x$. 4. If you have the graph of $f(x)$, then: - The graph of $-f(x)$ is obtained by taking each point $(x,y)$ on $f(x)$ and mapping it to $(x,-y)$. - The graph of $f(-x)$ is obtained by taking each point $(x,y)$ on $f(x)$ and mapping it to $(-x,y)$. - The graph of $-f(-x)$ is obtained by taking each point $(x,y)$ on $f(x)$ and mapping it to $(-x,-y)$. 5. Without a specific function $f(x)$, we cannot plot exact graphs, but these rules apply to any function. Final answer: The graphs of $-f(x)$, $f(-x)$, and $-f(-x)$ are reflections of the graph of $f(x)$ across the x-axis, y-axis, and both axes respectively.