A translation moves every point without changing the graph’s shape, a reflection flips the graph across an axis, and a stretch changes its vertical or horizontal scale. For transformations inside the function, the direction or scale factor is the opposite of what the algebra may initially suggest.
This is AHL 2.8: Transformations of graphs and composite transformations, so it is HL-only content and may appear on any GDC-active Math AI HL paper.
| Transformation | Equation from | Effect on the graph |
|---|---|---|
| Vertical translation | Moves up units if , or down if | |
| Horizontal translation | Moves right units if , or left if | |
| Reflection in the -axis | Each point ((x,y)) becomes ((x,-y)) | |
| Reflection in the -axis | Each point ((x,y)) becomes ((-x,y)) | |
| Vertical stretch | Multiplies every -coordinate by ; the -axis is invariant | |
| Horizontal stretch | Multiplies every -coordinate by ; the -axis is invariant |
The mechanism is easiest to see by tracking coordinates. If a point ((u,f(u))) lies on the original graph, then contains the point . Likewise, maps it to ((u/q,pf(u))). This coordinate view explains both “inside means inverse” rules: adding or multiplying the input changes the -coordinate oppositely, while operations outside act directly on the -coordinate. For negative or , the stretch also includes a reflection in the corresponding invariant axis.
For example, starting with , the graph of
is translated three units right, stretched vertically by scale factor , and translated one unit up. Its vertex moves from to .
A common misconception is that moves left because of the minus sign. It actually moves right: replacing affects horizontal coordinates inversely. Similarly, has horizontal scale factor , not .
Exam technique: When asked to describe composite transformations, state the transformation, direction or invariant axis, and scale factor. Apply them in the correct order because composite transformations may produce different results when reversed.