Function Transformations

Lesson

Once you know the parent functions, you can build new functions by transforming them — shifting, reflecting, or stretching the graph. The rules are mechanical once you see the pattern.

Starting from any parent $f(x)$, here’s what each operation does:

• $f(x) + k$ — shift up by $k$ (down if $k$ is negative).
• $f(x - h)$ — shift right by $h$. (Counterintuitive: minus inside moves the graph right.)
• $-f(x)$ — reflect across the $x$-axis (flip vertically).
• $f(-x)$ — reflect across the $y$-axis (flip horizontally).
• $a \cdot f(x)$ — stretch vertically by factor $a$ (compress if $|a| < 1$).

Multiple transformations combine: in $g(x) = a\cdot f(x - h) + k$, the inside $(x - h)$ shifts horizontally, the factor $a$ stretches and (if negative) reflects, and the $+ k$ shifts vertically.

To evaluate a transformed function at a specific $x$, just substitute and simplify.