Transformations: combining two

Order is the whole game

You already know the moves: translations, reflections and stretches. Exam questions love to chain two of them, and then the only real question is: which one happens first?

Sometimes it matters a lot. Take $y = 2f(x) - 3$ , and follow a point $(p, q)$ on the original graph $y = f(x)$ :

Stretch, then translate: double every height, then drop it by 3. Point $(p, q) \to (p, 2q - 3)$ ✓
Translate, then stretch: drop by 3, then double. Point $(p, q) \to (p, 2q - 6)$ ✗

Same two moves, different graph. So let's make the order automatic.

Quick reminder of the single moves ( $k$ is any number):

Equation	What happens to the graph of $y = f(x)$	Point $(p, q)$ goes to
$y = f(x) + k$	moves up $k$	$(p,\ q + k)$
$y = f(x + k)$	moves left $k$	$(p - k,\ q)$
$y = kf(x)$	heights multiplied by $k$ (stretch parallel to the $y$ -axis)	$(p,\ kq)$
$y = f(kx)$	$x$ -values divided by $k$ (stretch parallel to the $x$ -axis, factor $\tfrac1k$ )	$\left(\tfrac{p}{k},\ q\right)$
$y = -f(x)$	flipped in the $x$ -axis	$(p,\ -q)$
$y = f(-x)$	flipped in the $y$ -axis	$(-p,\ q)$

Notation: a translation is written as a column vector $\begin{pmatrix} h \\ k \end{pmatrix}$ . The top number is how far to move right (negative means left); the bottom number is how far to move up (negative means down). So $\begin{pmatrix} 0 \\ -3 \end{pmatrix}$ means "3 down, no sideways move".

Outside the bracket: follow the order of operations

For $y = af(x) + b$ , read it as a calculation done to the output $f(x)$ : multiply by $a$ , then add $b$ . Do the transformations in that same order:

Stretch parallel to the $y$ -axis with factor $a$ .
Then translate by $\begin{pmatrix} 0 \\ b \end{pmatrix}$ .

A minus sign counts as "multiply by $-1$ ": $y = 5 - f(x) = -f(x) + 5$ is a reflection in the $x$ -axis, then a translation by $\begin{pmatrix} 0 \\ 5 \end{pmatrix}$ .