Topic 1.12 Transformations of Functions: construct and analyze additive and multiplicative transformations (translations, dilations and reflections) of a function and their effect on the graph, domain and range.

What this topic is asking

The College Board (Topic 1.12) wants you to build and analyze transformations of a function: translations (shifts), dilations (stretches and compressions) and reflections. You must connect the change in the equation to the change in the graph, and determine how each transformation affects the domain and range.

Additive transformations: translations

The counterintuitive part is the horizontal shift: $f(x - 3)$ moves right, not left, because the input must be $3$ larger to produce the same output the original gave at the smaller input.

Multiplicative transformations: dilations and reflections

Vertical dilations multiply the outputs, so they scale the range. Horizontal dilations divide the inputs, so they scale the domain. A factor with absolute value greater than $1$ stretches vertically (or compresses horizontally), and a factor between $0$ and $1$ does the reverse.

Order and combining transformations

When several transformations combine, apply them consistently: horizontal changes happen inside the function (and act in the opposite sense), vertical changes happen outside (and act directly). For the range, run the original output interval through the vertical operations in order (dilate or reflect, then translate). For the domain, run the original input interval through the horizontal operations.

Why inside changes feel backwards

The horizontal rule trips almost everyone, so it is worth stating the reason once. In $f(x - h)$, the graph at the new input $x$ shows the original output that $f$ produced at $x - h$. To get the same output the original gave at input $0$, the new input must be $h$, so the feature has moved $h$ units to the right. The same logic explains why $f(bx)$ compresses by $\frac{1}{b}$: the input is scaled before $f$ acts, so the graph reaches each original output at a smaller input. Tying the rule to "what input is needed to reproduce a given output" turns a memorized reversal into something you can re-derive under pressure.

A second point is keeping vertical and horizontal effects separate. Vertical transformations change outputs and hence the range; horizontal transformations change inputs and hence the domain. Sorting each transformation into the vertical or horizontal bucket first, then applying it to the matching interval, prevents the common mistake of shifting the range when only the domain should move, and it makes combined transformations routine.

Try this

Q1. Describe the transformation taking $f(x)$ to $f(x) - 5$. [1 point]

• Cue. A vertical translation down by $5$ (outputs decrease by $5$).

Q2. The graph of $f$ is reflected across the $y$-axis. Which transformation produces this? [1 point]

• Cue. $f(-x)$: replacing $x$ with $-x$ reflects across the $y$-axis.