Reflections of graphs

Graphs can be reflected in either the \(x\) or \(y\) axes.

If \(f(x) = x^2\), then \(-f(x) = -(x^2)\).

This means that each of the \(y\) coordinates will have a sign change.

So \(y = 4\) would become \(y = -4\), and \(y = -1\) would become \(y = 1\) and so on.

This is what causes the reflection about the \(x\)-axis.

If \(f(x) = x^3\), then \(f(-x) = (-x)^3\).

This means that each of the \(x\) coordinates will have a sign change.

So \(x = 1\) would become \(x = -1\), and \(x = -2\) would become \(x = 2\) and so on.

This is what causes the reflection about the \(y\)-axis.

Question

If \(f(x)\) goes through the point \(\left(2, 4\right)\) then what is the equivalent point in the following functions?

\(y = f(x) + a \rightarrow\) translate up/down
\(y = f(x + a) \rightarrow\) translate left/right by –a
\(y = af(x) \rightarrow\) stretch the graph by multiplying the \(y\) coordinates
\(y = f(ax) \rightarrow\) compress the graph by dividing the \(x\) coordinates
\(y = -f(x) \rightarrow\) reflect in the \(x\)-axis
\(y = f(-x) \rightarrow\) reflect in the \(y\)-axis