Translations, Reflections, and Rotations on the Coordinate Plane

A transformation moves or resizes a figure according to a rule. On a coordinate plane, the rule changes each vertex's ordered pair. The most useful first step is to describe what happens to a point: a translation slides it, a reflection flips it, a rotation turns it, and a dilation scales its distance from a center. For a polygon, apply the same rule to every vertex and connect the new points in their original order.

Translations: slide by a fixed amount

A translation moves every point the same number of units horizontally and vertically. A rule such as (x, y) → (x + 4, y − 2) shifts the figure 4 units right and 2 units down. For point (−1, 5), the image is (−1 + 4, 5 − 2) = (3, 3). The x-change gives left or right; the y-change gives down or up.

A translation preserves lengths, angle measures, shape, and orientation. It changes location but not size. A sign error is common: adding a positive number to y moves up, while subtracting from y moves down. Check the result on graph paper or by counting grid squares from the original point.

Reflections: flip across a line

A reflection makes a mirror image across a line called the line of reflection. Reflecting across the x-axis changes the sign of y: (x, y) → (x, −y). Reflecting (3, −2) across the x-axis gives (3, 2). The point keeps its horizontal coordinate and moves the same distance to the opposite side of the x-axis.

Reflecting across the y-axis changes the sign of x: (x, y) → (−x, y). The point (−4, 1) becomes (4, 1). A reflection across the line y = x swaps the coordinates, (x, y) → (y, x), though a problem's permitted reflection line may be limited to an axis or a line parallel to an axis. Always read the stated line; different mirror lines create different coordinate rules.

A reflection preserves side lengths and angle measures but reverses orientation. For example, a triangle's image remains congruent to the original, yet the order of its vertices appears reversed when viewed around the shape. Points on the reflection line stay fixed; other points move to the opposite side at equal perpendicular distance.

Rotations about the origin

A rotation turns a figure around a center by a specified angle and direction. When the center is the origin, common counterclockwise rules are 90°: (x, y) → (−y, x); 180°: (x, y) → (−x, −y); and 270°: (x, y) → (y, −x). A 360° turn returns every point to its original location. Clockwise and counterclockwise turns of the same size use opposite directions.

For a 90° counterclockwise rotation, point (2, 5) becomes (−5, 2). Think of the rule as swapping coordinates, then changing one sign. For 180°, (2, 5) becomes (−2, −5), which is directly opposite the original across the origin. For 270° counterclockwise, (2, 5) becomes (5, −2), the same result as 90° clockwise.

These shortcut rules apply to rotations centered at the origin. If the center is another point, a direct sign-swap rule is not enough: translate the center to the origin, rotate, then translate back. If the question states an origin-centered rotation, do not add unnecessary steps.

Dilations: change size from a center

A dilation multiplies a point's coordinates by a scale factor when centered at the origin: (x, y) → (kx, ky). With k = 2, point (3, −1) becomes (6, −2); its distance from the origin doubles. A scale factor between 0 and 1 shrinks the figure toward the center. A positive scale factor preserves orientation; a negative factor also places the image on the opposite ray and can be understood with a half-turn.

A dilation preserves angle measures and shape, but generally changes side lengths and area. If the scale factor is k, side lengths are multiplied by |k| and areas by k². A dilation with scale factor 1 leaves the figure unchanged. Unlike translation, reflection, and rotation, a dilation is not usually a rigid transformation because it changes distances.

Transform every vertex consistently

Suppose triangle ABC has vertices A(1, 1), B(4, 1), and C(2, 3), and the rule is (x, y) → (x − 2, y + 1). Apply it separately to A, B, and C: A′(−1, 2), B′(2, 2), and C′(0, 4). Connect A′ to B′ to C′ in the same vertex order. If you transform only one point or mislabel an image vertex, the graph will not represent the whole transformed triangle.

A compact table with original and image coordinates reduces errors. Write the transformation rule above the table, substitute both coordinates, and check each result. For a multi-step problem, apply transformations in the order given; generally, changing the order can produce a different final image.

Recognize rigid transformations

Translations, reflections, and rotations are rigid transformations: they preserve all distances and angle measures, so the image is congruent to the original. A dilation preserves angle measures and shape but changes size unless its scale factor has magnitude 1. If a question asks which transformation preserves distance, choose among the rigid motions rather than a size-changing dilation.

Some figures map onto themselves after a transformation because of symmetry. A square maps onto itself after a 90° rotation about its center, and it maps onto itself after reflection across a line through opposite vertices or side midpoints. The image may occupy the same region even though vertices trade places. Compare the whole figure, not just one labeled vertex.

Common errors and a check routine

Coordinate rules are easy to confuse because all use the same pair of numbers. Translation adds or subtracts fixed amounts. Reflection changes a sign or swaps coordinates depending on the mirror line. Rotation swaps and changes signs according to its angle and direction. Dilation multiplies by a scale factor. Identify the motion first, then use only its rule.

1. Name the transformation, its center or line, and its direction or scale factor.
2. Write the coordinate rule before substituting a point.
3. Apply the rule to every vertex and label each image point with a prime.
4. Plot the image or compare distances and angles to check the result.
5. For a sequence, apply each rule in the order stated.

Exam takeaway

Translate by adding fixed coordinate changes; reflect by flipping across the specified line; rotate using the angle-and-direction rule; dilate by multiplying distances from the center by the scale factor. Transform every vertex, preserve labels, and use distance and angle checks to confirm whether size should stay the same.