Reflections, rotations, and translations move points on the coordinate plane by fixed rules—memorize the sign flips for reflections, the (x, y) → swaps for rotations about the origin, and the slide-by-(h, k) pattern for translations. These are geometry transformations, not the same as function transformations like f(x - 2).
Reflections
Reflecting over the x-axis: (x, y) → (x, -y). Over the y-axis: (x, y) → (-x, y). Over the line y = x: swap coordinates, (x, y) → (y, x). Example: (3, -2) reflected over the x-axis is (3, 2); over y = x it becomes (-2, 3).
Rotations about the origin
- 90° counterclockwise: (x, y) → (-y, x)
- 180°: (x, y) → (-x, -y)
- 270° counterclockwise (or 90° clockwise): (x, y) → (y, -x)
Example: rotate (4, 1) by 90° CCW to get (-1, 4). A quick sketch on your test booklet prevents sign errors.
Translations
Slide every point by (h, k): (x, y) → (x + h, y + k). A translation 3 units right and 2 units down means h = 3, k = -2, so (1, 5) → (4, 3). Translations preserve shape and orientation—only position changes.
Not the same as function transformations
Moving the point (2, 3) two units left gives (0, 3). Shifting the graph of f(x) two units left is f(x + 2)—a different operation on a different object. If you mix them up, read ACT function transformations alongside this post.
Where to practice
These rules live inside coordinate geometry on the ACT. For how much geometry to expect overall, see how much geometry is on the ACT. Keep a formula sheet with the rotation rules—our ACT Math formulas guide is a good place to consolidate them.
Start practicing
Start with a free diagnostic, then drill your weak spots with 15-question quizzes and track how you're doing across Reading, English, and Math. Compare plans whenever you're ready to go further.
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