Perform sequences of transformations

A sequence of transformations applies two or more translations, reflections, rotations, and dilations in a specified order, using coordinate rules to determine each intermediate and final image of a point or figure. The learner understands that transformations generally do not commute: reversing their order can produce a different image, while the resulting coordinates preserve or change lengths and angles according to whether the sequence establishes congruence or similarity. Matrix methods and generalized composition formulas are beyond this scope.

Detailed Explanation: Perform sequences of transformations

To perform a sequence of transformations, apply the rules one at a time and in the order given. The result from one step becomes the starting point for the next step.

Useful coordinate rules:

Translate right $a$ and up $b$ : $(x,y)\rightarrow(x+a,y+b)$
Reflect across the $y$ -axis: $(x,y)\rightarrow(-x,y)$
Rotate $90^\circ$ counterclockwise about the origin: $(x,y)\rightarrow(-y,x)$

Example: Point $P(-2,3)$ is translated $4$ units right and $1$ unit down, then reflected across the $y$ -axis, and finally rotated $90^\circ$ counterclockwise about the origin. Find the final image of $P$ .

Step 1: Translate

Moving $4$ units right adds $4$ to the $x$ -coordinate. Moving $1$ unit down subtracts $1$ from the $y$ -coordinate:

(-2,3)\rightarrow(-2+4,3-1)=(2,2)

After the translation, the point is $P'(2,2)$ .

Step 2: Reflect across the $y$ -axis

A reflection across the $y$ -axis changes the sign of the $x$ -coordinate but keeps the $y$ -coordinate the same:

(2,2)\rightarrow(-2,2)

After the reflection, the point is $P''(-2,2)$ .

Step 3: Rotate $90^\circ$ counterclockwise

For this rotation, switch the coordinates and change the sign of the original $y$ -coordinate:

(x,y)\rightarrow(-y,x)

Apply the rule to $(-2,2)$ :

(-2,2)\rightarrow(-2,-2)

Therefore, the final image is

\boxed{P'''(-2,-2)}

The order matters. Each transformation uses the point produced by the previous step, so reversing the steps can give a different final location.

Learn by doing: Perform sequences of transformations

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Multiple Transformations - Grid Image & Transformations to Final Image

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