5.5 Transformations on the Coordinate Plane

Why Transformations Matter on the SHSAT

Transformations describe how figures move on the coordinate plane. The SHSAT tests four transformation types — translations, reflections, rotations, and dilations — in both rule-based form ("If (x, y) → (x + 3, y - 2), what are the new coordinates of point P?") and graph-based form ("Triangle ABC is reflected across the y-axis; which point represents the image of B?"). Transformations also appear in similarity questions, because a dilation is the geometric foundation of scale drawings and similar figures.

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Translations

A translation slides every point of a figure the same distance in the same direction. The image is congruent to the preimage.

Coordinate Rule

A translation of a units horizontally and b units vertically maps every point (x, y) to:

$(x, y) \rightarrow (x + a, y + b)$

Example: Point A is at (3, -2). After the translation (x, y) → (x - 4, y + 5), the image is A' = (3 - 4, -2 + 5) = (-1, 3).

SHSAT Cues

Look for language such as "shifted 3 units left and 2 units up." Left means subtract from x; right means add to x; up means add to y; down means subtract from y.

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Reflections

A reflection flips a figure across a line of reflection, producing a mirror image. The image is congruent to the preimage.

Common Reflection Rules

Example: Reflect point B(-3, 5) across the y-axis. The rule (x, y) → (-x, y) gives B'(3, 5).

Example: Reflect point C(4, 1) across the line y = x. The rule (x, y) → (y, x) gives C'(1, 4).

Distance-to-Line Check

Every point and its reflected image are equidistant from the line of reflection, and the segment connecting them is perpendicular to that line. Use this check on graph-based items to confirm your answer.

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Rotations

A rotation turns a figure about a fixed point by a given angle. The image is congruent to the preimage. SHSAT rotations are typically 90°, 180°, or 270° counterclockwise about the origin.

Rotation Rules About the Origin

Example: Rotate point D(2, 5) by 90° counterclockwise about the origin. Using (x, y) → (-y, x), the image is D'(-5, 2).

Clockwise Shortcuts

A 90° clockwise rotation is equivalent to a 270° counterclockwise rotation: (x, y) → (y, -x). A 180° rotation is the same in either direction.

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Dilations

A dilation enlarges or reduces a figure by a scale factor k with respect to a center point (usually the origin). A dilation produces a similar figure: angles are preserved, but side lengths and area change.

Coordinate Rule (center at origin)

$(x, y) \rightarrow (kx, ky)$

Effect on Measurements

• Side lengths scale by |k|.
• Area scales by k^2.
• Perimeter scales by |k|.
• Angle measures are unchanged.

Example: A triangle has vertices (2, 4), (6, 4), and (2, 8). After a dilation with scale factor 3 about the origin, the image has vertices (6, 12), (18, 12), and (6, 24). Side lengths triple; the area is 9 times the original.

Scale Factor Interpretation

• k > 1: enlargement.
• 0 < k < 1: reduction.
• k = 1: identity (no change).
• k < 0: enlargement or reduction plus a 180° rotation about the center.

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Congruence vs. Similarity

Translations, reflections, and rotations are rigid motions; they produce congruent figures. Dilations are non-rigid; they produce similar figures. A sequence of rigid motions followed by a dilation produces a figure similar to the original.

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Common SHSAT Traps

• Sign flips on rotations: A 90° counterclockwise rotation maps (x, y) to (-y, x), not (-x, y). Memorize the rule, or sketch the unit circle to derive it.
• Confusing reflection lines: Reflection across the x-axis flips the y-coordinate; reflection across the y-axis flips the x-coordinate. Mixing these up is the most common reflection error.
• Dilating from a non-origin center: When the center is (h, k), the rule is (x, y) → (h + k(x - h), k + k(y - k)). SHSAT items usually keep the center at the origin, but verify before applying the simple (kx, ky) rule.
• Area scaling: Many students multiply area by k instead of by k^2. Always square the scale factor for area and cube it for volume.

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Worked SHSAT-Style Example

Triangle ABC has vertices A(1, 2), B(4, 2), and C(1, 6). The triangle is reflected across the y-axis and then translated 3 units right and 1 unit down. What are the coordinates of the final image of vertex C?

Step 1 — Reflect C(1, 6) across the y-axis: (x, y) → (-x, y) gives C'(-1, 6). Step 2 — Translate by (x + 3, y - 1): C''(-1 + 3, 6 - 1) = (2, 5). The final image of C is (2, 5).