Graphs & Transformations of Functions

Transformations describe how the graph of $y = f(x)$ moves or reshapes when the function is altered. The single rule that unlocks them all: changes outside $f$ affect the graph vertically (as you’d expect); changes inside $f$ affect it horizontally (the opposite of what you’d expect).

Translations

• $y = f(x) + a$: shifts the graph up by $a$ (outside $f$, vertical, intuitive).
• $y = f(x + a)$: shifts the graph left by $a$ (inside $f$, horizontal, reversed, $+a$ moves left, not right).

The inside-is-reversed behaviour is the most common source of error. $f(x - 3)$ moves the graph right by 3.

Reflections

• $y = -f(x)$: reflects in the $x$-axis (the outputs flip sign).
• $y = f(-x)$: reflects in the $y$-axis (the inputs flip sign).

Stretches

• $y = a\,f(x)$: a vertical stretch, scale factor $a$ (each $y$-value multiplied by $a$).
• $y = f(ax)$: a horizontal stretch, scale factor $\frac{1}{a}$ (again the inside behaves inversely, $f(2x)$ compresses by a factor of 2).

A worked check

The graph of $y = f(x)$ has a maximum at $(2, 5)$. State the coordinates of the maximum of $y = f(x - 1) + 3$. $f(x - 1)$ shifts right 1: the $x$-coordinate becomes $2 + 1 = 3$. The $+3$ shifts up 3: the $y$-coordinate becomes $5 + 3 = 8$. Maximum at $(3, 8)$.

Track a single known point (like the vertex) through each transformation, it’s the reliable way to avoid sign confusion.

Common mistakes

• Moving $f(x + a)$ to the right instead of the left (inside is reversed).
• Confusing a vertical stretch $af(x)$ with a horizontal one $f(ax)$.
• Applying a horizontal stretch factor of $a$ instead of $\frac{1}{a}$.

These transformations underpin sketching in modulus graphs and trig graphs. Full topic context: Functions notes.