Apply a Reflection before a Translation

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Reflect (2,3) across the x-axis, then translate by (1,2). Candidate answers: (A) (-1,5) (B) (3,-1) (C) (3,-5) (D) (3,5).
• Independent check inputs: Reflect (3,4) across the x-axis, then translate by (1,2). Candidate answers: (A) (-2,6) (B) (4,-2) (C) (4,-6) (D) (4,6).

Learn the idea

Translations, reflections and rotations preserve distances and angle measures. Coordinate rules describe these moves exactly, and a sequence of moves must be applied in its stated order to determine the final image. Composing transformations means applying each move to the result of the previous one. A translation following a reflection acts on the reflected coordinates, not on the original point.

Here is the method on the supplied teaching case. First obtain (2,−3); then add the translation to get (2+1,−3+2). Reversing the operations would generally produce a different point.

The order is part of the instructions; these moves need not commute. Record the intermediate point so a later step does not accidentally reuse the original coordinates or reflect the translation itself.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Reflect (9,10) across the x-axis, then translate by (1,2). Candidate answers: (A) (-8,12) (B) (10,-12) (C) (10,-8) (D) (10,12). Show the mathematical reason, not only the selected answer. Expected result: (10,-8). First obtain (9,−10); then add the translation to get (9+1,−10+2). Reversing the operations would generally produce a different point.

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