For $y=-f(x)$, we reflect $y=f(x)$ in the $x$-axis.
For $y=f(-x)$, we reflect $y=f(x)$ in the $y$-axis.
Example 1
Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=-f(x)$.
\( \begin{align} \displaystyle
f(x) &= x^3-4x^2+4x \\
&= x(x^2-4x + 4) \\
&= x(x-2)^2
\end{align} \)
Example 2
Consider $f(x)=x^3-4x^2+4x$. On the same axes, sketch the graphs of $y=f(x)$ and $y=f(-x)$.
\( \begin{align} \displaystyle
f(x) &= x^3-4x^2+4x \\
&= x(x^2-4x + 4) \\
&= x(x-2)^2
\end{align} \)
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 $f(x)$.
$-f(x)$ means that the graph is to reflect $y=f(x)$ in the $x$-axis, which the sign of $y$-value changes.
Thus the $y$-value $0$ was transformed to $0$.
Therefore the point $(3,0)$ was transformed from $(3,0)$.
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 $g(x)$.
$-f(x)$ means that the graph is to reflect $y=f(x)$ in the $x$-axis, which the sign of $y$-value changes.
Thus the $y$-value $-1$ was transformed from $1$.
Therefore the point $(7,-1)$ was transformed from $(7,1)$.
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