Family Overview

In Plain Language

A transformation builds a new function from a parent $f$ by shifting, stretching/compressing, or reflecting its graph.

• Additive: $g(x)=f(x)+k$ moves the graph vertically by $k$. $g(x)=f(x+h)$ moves it horizontally by $-h$ (left if $h>0$, right if $h<0$).
• Multiplicative: $g(x)=a f(x)$ stretches/compresses vertically by $|a|$ (and reflects over the $x$-axis if $a<0$). $g(x)=f(bx)$ stretches/compresses horizontally by $1/|b|$ (and reflects over the $y$-axis if $b<0$).
• Combinations stack these moves. Domain and range of the new function may differ from the parent.

Why It Matters

AP Precalculus asks students to build $g$ from $f$, describe the geometric effect, map key points, and track domain/range changes—skills used later in modeling and in every function family.

How Families Can Help

• Ask: “Is the change inside $f(\ldots)$ or outside?”
• Practice: “Inside affects $x$ (horizontal); outside affects $y$ (vertical).”
• Map one known point through each step of a combo.

What Not to Do

• Do not reverse horizontal shift direction ($f(x+h)$ is left when $h>0$).
• Do not treat $|a|$ stretch the same as the $b$ horizontal factor ($1/|b|$).
• Do not ignore reflections when $a<0$ or $b<0$.
• Do not assume domain/range stay identical to the parent.