Determine whether the graph of $y = |x − 4|$ is symmetric with respect to the origin, the $x$-axis, or the $y$-axis.
A. not symmetric with respect to the $x$-axis, not symmetric with respect to the $y$-axis, and not symmetric with respect to the origin
B. symmetric with respect to the $x$-axis only
C. symmetric with respect to the $y$-axis only
D. symmetric with respect to the origin only
I assume that the answer is B but I am not sure if my choice is correct or not. Please help.
You should get the answer quickly if you have the graph of the curve. But assuming that the curve is not available (or it is too complicated to graph), then we can use the definition of the symmetries.
A curve is symmetric with respect to the $y$-axis if the equation of the curve is unchanged under the mapping $x \mapsto -x$; this mapping reflects all points across the $y$-axis.
Similar, a curve is symmetric with respect to the $x$-axis if the equation of the curve is unchanged under the mapping $y \mapsto -y$.
Combining the two, a curve is symmetric with respect to the origin if the equation of the curve is unchanged under the mapping $(x,y) \mapsto (-x,-y)$.
Now to the example \begin{equation} y = |x-4|. \tag{$*$} \end{equation} If we perform the mapping $x \mapsto -x$ to $(*)$, then we get $y=|-x-4|$, which is different from $(*)$. Therefore $(*)$ is not symmetric with respect to the $y$-axis.
I shall let you determine if $(*)$ has the other two types of symmetries.