Compose and undo transformations

Compose replacements in the stated order

Work with the current equation at each step. Write the intermediate equation before applying the next transformation; applying every change to the original equation can give the wrong result.

Changes to different coordinates

In Example 7, horizontal scaling and vertical translation affect different coordinates. Either order gives $f(2x)+1$:

• Replace x by $2x$, then y by $y-1$.
• Or replace y by $y-1$, then x by $2x$.

These are the two sequences compared in the Teacher notes.

Changes to the same coordinate

Order matters when a later scaling also scales an earlier translation. Example 8 reaches $f(2x+1)$ in either of these ways:

1. Scale horizontally by $1/2$ to get $f(2x)$. Translate left by $1/2$ to get $f(2(x+1/2))=f(2x+1)$.
2. Translate left by 1 to get $f(x+1)$. Scale horizontally by $1/2$ to get $f(2x+1)$.

The translation distances differ. Swapping the order while keeping the same distances would change the final graph.

Example 9 makes the same point vertically: reach $2f(x)+1$ by scaling by 2 then translating up 1, or by translating up $1/2$ then scaling by 2. Keep the intermediate replacement equations so you can see which terms the scaling affects.

Choose a sequence, then verify it

The source recommends horizontal translation, horizontal scaling, vertical scaling, then vertical translation. Its mnemonic $T_xS_xS_yT_y$ is a convenient guide, not a restriction on other valid sequences. Check the resulting equation, especially if you change the order.

Examples 10–11 show each replacement. Completing the square gives $5(x-2)^2-4$; the transformed cubic has exact intercepts $(0,-10)$ and $(2+\sqrt[3]{12},0)$. In Example 11, “stretch” by a factor of $1/2$ means halve vertical distances.

Undo a sequence

Reverse the order and invert each transformation. If you first scale vertically by 2 and then move up 1, undo the move first, then divide the height by 2.

Example 12 undoes C, B, A to obtain $y=\ln(-x)$. It also shows the Teacher’s alternative point map $(p,q)\mapsto(p/2+3,-q)$; the real-domain check is an independently authored supplement.

Check before the examples: after scaling vertically by 2 and moving up 1, an original height of 3 becomes 7. Undoing those steps gives $(7-1)/2=3$, not $7/2-1$.

Self-Practice 4 includes all three two-step answers printed in both copies. Horizontal translation by 2 before scaling can correspond to translation by $2/3$ after scaling. The verification equations are authored explanations of the printed answers.