Reflections of graphs
Graphs can be reflected in either the \(x\) or \(y\) axes.
Reflections in the x-axis
If \(f(x) = x^2\), then \(-f(x) = -(x^2)\).
Reflections in the y-axis
If \(f(x) = x^3\), then \(f(-x) = (-x)^3\).
Question
If \(f(x)\) goes through the point (2, 4) then what is the equivalent point in the following functions?
- \(f(x) + 1\)
- \(f(x + 2)\)
- \(-f(x)\)
- \(f(-x)\)
- \(f(x) + 1\) goes through the point (2, 5).
- \(f(x + 2)\) goes through the point (0, 4).
- \(-f(x)\) goes through the point (2, -4).
- \(f(-x)\) goes through the point (-2, 4).
Remember:
- \(y = f(x) + a \rightarrow\) translate up/down by the vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\)
- \(y = f(x + a) \rightarrow\) translate left/right by the vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\)
- \(y = -f(x) \rightarrow\) reflect in the \(x\)-axis
- \(y = f(-x) \rightarrow\) reflect in the \(y\)-axis