Graph transformations

The idea

Changes outside the function act on $y$ as you expect; changes inside act on $x$ the opposite way. $y = f(x) + a$ moves up $a$; $y = f(x + a)$ moves left $a$; $y = af(x)$ stretches vertically by $a$; $y = f(ax)$ stretches horizontally by $\frac{1}{a}$.

1. Decide whether the change is inside or outside the brackets.
2. Outside: change the $y$-coordinates. Inside: change the $x$-coordinates, inverted.
3. Move each key point, asymptote and intercept.

Watch out $f(2x)$ halves the $x$-coordinates; it does not double them.