Advanced shifting functions

Several transformations can be combined into one form: $g(x) = a\,f(x - h) + k$.
Each piece does a separate job, and reading them one at a time keeps it simple.
The changes inside the function affect the graph horizontally, and the changes outside affect it vertically.

The $(x - h)$ inside causes a horizontal shift, and its sign is opposite to what it looks like.
So $f(x - 3)$ moves the graph $3$ units to the right, while $f(x + 3)$ moves it $3$ units to the left.

The $a$ in front is a vertical stretch, and a negative $a$ also flips the graph over the $x$-axis.
The $+ k$ at the end is a vertical shift: $+k$ moves the graph up by $k$, and $-k$ moves it down.
For example, $2f(x) - 1$ makes the graph twice as tall and then lowers it by $1$.

For a parabola, this form is especially useful and is called vertex form: $y = a(x - h)^2 + k$.
Its vertex sits exactly at $(h, k)$, so $y = (x - 2)^2 + 1$ has its vertex at $(2, 1)$.

When several changes appear together, apply them in order.
Take $g(x) = -f(x + 1) - 4$: the $+1$ inside shifts $f$ left $1$, the minus sign flips it over the $x$-axis, and the $-4$ lowers it by $4$.
Handling one piece at a time avoids mixing up the directions.

How does $g(x) = f(x - 3) + 2$ transform the graph of $f(x)$?
The $-3$ inside shifts the graph $3$ units to the right, and the $+2$ outside shifts it $2$ units up.
So $g$ is $f$ moved right $3$ and up $2$.

Where is the vertex of the parabola $y = 2(x - 1)^2 - 4$?
In vertex form $y = a(x - h)^2 + k$, the vertex is at $(h, k)$.
Here $h = 1$ and $k = -4$, so the vertex is at $(1, -4)$.

The graph below shows a function $f(x)$ and its transformation $g(x) = -f(x + 2) - 1$.

How does $g$ transform $f(x)$? \ The $+2$ inside shifts $f$ left $2$, the minus sign flips it over the $x$-axis, and the $-1$ lowers it by $1$. \ So $g$ is $f$ reflected over the $x$-axis, shifted left $2$, and down $1$.