Transformations

A transformation is simply moving or reshaping a graph you already know: sliding it, stretching it or flipping it. If you know what $\sqrt x$ looks like, you know what $-2\sqrt{x + 1} + 3$ looks like without plotting a single point. The formula $a\,f(x - h) + k$ has three dials: $a$ (stretch it taller or flatter, and flip it if negative), $h$ (slide it left or right) and $k$ (slide it up or down). That's all there is. It's the same idea you saw with the parabola's vertex form, now for any graph.

Outside changes are easy. Adding $k$ to every output just raises every point by $k$, and multiplying every output by $a$ stretches every height by $a$. They do what they look like.

A common mistake. Since $+k$ moves the graph up, it's tempting to think $f(x - 2)$ moves it left, toward the minus. It moves right. Think about when the new graph reaches the value $f(0)$: that happens when $x - 2 = 0$, so at $x = 2$. Everything happens 2 units later, so the graph slides right. Inside changes act on the input, so they work backwards from what you'd guess.

Order. Do stretches and reflections before shifts. A stretch scales distances from the axis, so if you shift first, the stretch scales the shift too.

Animators and game engines move, scale and mirror sprites with exactly these operations. A delay in a signal is a horizontal shift; an amplifier is a vertical stretch.