Describe the effect of dilations, translations, rotations and reflections on figures using coordinates

Learning goal: Write the coordinate rule for translations, reflections across an axis, rotations about the origin and dilations centered at the origin, apply each rule to every vertex of a figure, and describe in words what the transformation does to the figure's position, size, orientation and shape.

Why coordinates. A drawing shows roughly what a transformation does; coordinates say exactly. Each transformation below has a rule of the form (x, y) → (something, something). Apply the rule to every vertex of a polygon, then connect the image vertices in the same order. The rules apply to points, so a figure with five vertices needs the rule applied five times.

Translation by a units horizontally and b units vertically: (x, y) → (x + a, y + b). Positive a moves right, negative a moves left; positive b moves up, negative b moves down. Effect: the image is congruent to the pre-image, in the same orientation, with every side still parallel to its original, just moved.

Reflection across the x-axis: (x, y) → (x, −y). The x-coordinate is unchanged because the point moves straight up or down onto the perpendicular through its own x; the y-coordinate changes sign because the image is the same distance from the axis on the other side. Reflection across the y-axis: (x, y) → (−x, y), for the same reason with the roles swapped. Effect: congruent image, orientation reversed. These two rules are only for the axes; reflecting across some other line has a different rule.

Rotation about the origin. A quarter turn counterclockwise takes (1, 0) to (0, 1) and (0, 1) to (−1, 0). The box with corners at the origin and (x, y) turns with the point, so 90° counterclockwise is (x, y) → (−y, x). Two quarter turns make a half turn: 180° is (x, y) → (−x, −y), the same in either direction. 90° clockwise is a quarter turn the other way, (x, y) → (y, −x), which is also 270° counterclockwise. Effect: congruent image, same orientation, turned. These rules are for the origin as center; a rotation about another point has a different rule.

Dilation centered at the origin with scale factor k (k positive): (x, y) → (kx, ky). Each point moves along the ray from the origin through it, to k times its original distance from the origin. If k is greater than 1 the image is an enlargement; if k is between 0 and 1 it is a reduction; k = 1 leaves everything fixed. Effect: the image has the same shape and the same angles, every length is multiplied by k, and every area is multiplied by k². A dilation with k ≠ 1 is not a rigid motion, because it changes lengths; its image is similar to the pre-image, not congruent. With k = 1 the dilation is the identity: nothing moves, and the image is the pre-image itself. The center itself does not move, so a figure not containing the origin also moves away from or toward the origin as it grows or shrinks.

A worked model. Triangle ABC has A(2, 1), B(6, 1), C(3, 4). Side AB is horizontal, length 4. Apply each rule to all three vertices.

Translation (x, y) → (x + 2, y − 3): A′(4, −2), B′(8, −2), C′(5, 1). Check C: 3 + 2 = 5 and 4 − 3 = 1. A′B′ is still horizontal with length 8 − 4 = 4.

Reflection across the x-axis (x, −y): A′(2, −1), B′(6, −1), C′(3, −4). The triangle now hangs below the axis; A′B′ has length 4.

Reflection across the y-axis (−x, y): A′(−2, 1), B′(−6, 1), C′(−3, 4). Note that B′ is now the leftmost point: the reflection reversed left and right.

Rotation 90° counterclockwise (−y, x): A′(−1, 2), B′(−1, 6), C′(−4, 3). Check C(3, 4): its distance from the origin is 5, the diagonal of a 3-by-4 box; C′(−4, 3) is the diagonal of a 4-by-3 box, also 5, and its direction has turned a quarter turn. A′B′ is now vertical, from height 2 to height 6, length 4.

Rotation 180° (−x, −y): A′(−2, −1), B′(−6, −1), C′(−3, −4).

Dilation, center the origin, k = 2: (2x, 2y): A′(4, 2), B′(12, 2), C′(6, 8). A′B′ has length 12 − 4 = 8, which is 2 × 4. A is √5 units from the origin (a 2-by-1 box) and A′ is √20 = 2√5 units away: twice as far, on the same ray.

Dilation, k = 1/2: A′(1, 0.5), B′(3, 0.5), C′(1.5, 2). A′B′ has length 2, half of 4.

Describing the effect in words. Coordinates are the evidence; the description is the conclusion. For the translation: "a congruent triangle, 2 units right and 3 units down, same orientation." For the 90° rotation: "a congruent triangle turned a quarter turn counterclockwise about the origin; the horizontal side became vertical." For the dilation by 2: "a similar triangle with every side doubled, every angle unchanged, and every vertex twice as far from the origin." Say what changed and what did not.

Recognizing a rule from coordinates. If every image coordinate is the pre-image coordinate plus the same constants, it is a translation. If one coordinate is unchanged and the other has changed sign, it is a reflection across an axis. If both changed sign, a 180° rotation. If the coordinates swapped places with one sign change, a 90° rotation; look at where (1, 0) would go to decide the direction. If both coordinates are multiplied by the same positive number, a dilation centered at the origin.

Misconceptions. The commonest error is mixing up the two 90° rules. Test the rule you wrote on the point (1, 0): counterclockwise must send it to (0, 1), clockwise to (0, −1). A second error is applying the rule to some vertices and copying the rest by eye; the image will be a slightly wrong shape. A third is treating a dilation as a slide: the image of a dilation is a different size, and the points nearer the origin move less than the points farther away.

Worked dilation of a rectangle: Rectangle with vertices (1, 1), (3, 1), (3, 2), (1, 2) is dilated from the origin with k = 3. Rule (3x, 3y): (3, 3), (9, 3), (9, 6), (3, 6). The 2-unit side became 9 − 3 = 6 units and the 1-unit side became 6 − 3 = 3 units, both multiplied by 3. The area went from 2 × 1 = 2 square units to 6 × 3 = 18 square units, multiplied by 3² = 9. The angles are still right angles.

Worked rotations of one point: Rotate (−2, 5) about the origin. 90° clockwise, (y, −x): (5, 2). 90° counterclockwise, (−y, x): (−5, −2). 180°, (−x, −y): (2, −5). All four points are the same distance from the origin, the diagonal of a 2-by-5 box, which is what a rotation about the origin must keep.

Worked identification: Pre-image (1, 2), (4, 2), (4, 6). Image 1: (−1, 2), (−4, 2), (−4, 6). Each y is unchanged and each x has changed sign: a reflection across the y-axis. Image 2: (2, 4), (8, 4), (8, 12). Each coordinate has been doubled: a dilation centered at the origin with k = 2; the vertical side grew from 4 to 8. Image 3: (−2, 1), (−2, 4), (−6, 4). Each (x, y) became (−y, x): a 90° counterclockwise rotation; check (1, 2) → (−2, 1).

Worked sequence: Apply (x, y) → (x − 5, y + 2) and then reflect across the x-axis to the point (3, 1). First (3 − 5, 1 + 2) = (−2, 3), then (−2, −3). Doing the reflection first gives (3, −1) and then (−2, 1), a different point, so the order must be stated.

Recording task: Make a table with the vertices A(2, 1), B(6, 1), C(3, 4) as its column headings and one row for each of the seven rules in the lesson: translation (x + 2, y − 3), reflection across the x-axis, reflection across the y-axis, rotation 90° counterclockwise, rotation 180°, dilation k = 2 and dilation k = 1/2. Fill every cell by applying the rule, then plot two of the rows on a grid and check by eye that the drawn image matches the description in words (moved, flipped, turned or resized).

Independent performance: Choose a quadrilateral with vertices on grid points, none on either axis, and with at least one side that is neither horizontal nor vertical. Apply all seven rules from the recording task and, for each, compute the length of one slanted side (neither horizontal nor vertical) before and after from its run and rise, and state in one sentence the effect on position, size and orientation. Success criteria: every image coordinate is correct when a partner tests the rule on one vertex; the side length is unchanged for the five rigid motions and multiplied by exactly k for the two dilations; the 90° counterclockwise rule is confirmed by testing it on (1, 0), which must go to (0, 1), and the clockwise rule (y, −x) is written beside it for contrast; and the sentences use the words congruent or similar correctly.