Reflections $y=-f(x)$ and $y=f(-x)$

For $y=-f(x)$, we reflect $y=f(x)$ in the $x$-axis. Reflection of Graphs
For $y=f(-x)$, we reflect $y=f(x)$ in the $y$-axis. Reflection of Graphs

Example 1

Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=-f(x)$.

Example 2

Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=f(-x)$.

Example 3

The function $y=f(x)$ is transformed to $g(x)=-f(x)$. Find the point on $g(x)$ corresponding to the point $(3,0)$ on $f(x)$.

Example 4

The function $y=f(x)$ is transformed to $g(x)=-f(x)$. Find the point on $f(x)$ that has been transformed to the point $(7,-1)$ on $g(x)$.