Transformations of graphs

Graphs can be transformed by translating, stretching, or reflecting them. These transformations follow predictable rules.

Translations:

• $f(x) + a$ — moves the graph up by $a$
• $f(x + a)$ — moves the graph left by $a$ (inside the function affects $x$ in reverse)

Stretches:

• $af(x)$ — stretches vertically by scale factor $a$
• $f(ax)$ — stretches horizontally by scale factor $\frac{1}{a}$

Reflections:

• $-f(x)$ — reflects in the $x$-axis
• $f(-x)$ — reflects in the $y$-axis

Example: Starting from $y = \sin x$: $$y = \sin(x + 30°) \quad \text{is a translation of } \begin{pmatrix} -30 \\\\ 0 \end{pmatrix}$$

A common trick: write transformations as vectors $\begin{pmatrix} a \\ b \end{pmatrix}$ for translations.

Common error: thinking $f(x+2)$ moves the graph right — it moves left. The rule is counterintuitive.