# Reflections of Graphs

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 $(7,1)$.