Transformations, Reciprocal and Rational Functions

Once you know the graph of a base function $y = f(x)$, you can produce whole families of related graphs by transformations. There are two categories: translations (slides), and stretches and reflections (scalings). The crucial rule to internalize is that changes applied outside the function affect the output ($y$, vertical) intuitively, while changes applied inside the function affect the input ($x$, horizontal) in the opposite way to what you might expect.

Translations

• $y = f(x) + k$ shifts the graph up by $k$ (down if $k < 0$). This is a vertical translation by $\begin{pmatrix} 0 \ k \end{pmatrix}$.
• $y = f(x - h)$ shifts the graph right by $h$ (left if $h < 0$). This is a horizontal translation by $\begin{pmatrix} h \ 0 \end{pmatrix}$. Note the counterintuitive sign: $f(x-3)$ moves righ