This grade 8 transformations lesson explains translations, reflections and rotations in one printable page, with definitions, a method and a worked example.
Translations, reflections and rotations are rigid motions: they move a figure without changing any side length or angle measure. On the coordinate plane a translation adds the same numbers to every point, a reflection changes a sign, and a rotation about the origin swaps the coordinates and changes signs.
Grade 8 Transformations Lesson: key facts
- Rigid motions keep segment lengths, angle measures and parallel lines, so the image is congruent to the pre-image.
- Translation: (x, y) becomes (x + a, y + b).
- Reflection across the x-axis: (x, -y). Across the y-axis: (-x, y).
- Rotation about the origin, counterclockwise: 90 degrees (-y, x), 180 degrees (-x, -y), 270 degrees (y, -x).
- The image of point A is written A prime.
What is a rigid motion?
A rigid motion moves a figure without changing its size or shape. The three kinds are translations, reflections and rotations. The original figure is the pre-image, and the moved figure is the image. The image of point \(A\) is named \(A^{\prime}\).
Translation: a slide of every point the same distance in the same direction.
Reflection: a flip across a line, such as an axis.
Rotation: a turn about a point. Here the point is the origin and the turn is counterclockwise.
Every rigid motion keeps lines as lines, keeps the length of every segment, keeps the measure of every angle, and keeps parallel lines parallel. So the image is congruent to the pre-image.
How do you translate a figure on the coordinate plane?
Add the same numbers to every point. Moving 6 right and 3 down changes \((x,y)\) to \((x+6,y-3)\). The point \(A(-5,1)\) lands on \(A^{\prime}(1,-2)\).
How do you reflect a figure across an axis?
A reflection across the \(x\)-axis changes the sign of the \(y\)-coordinate. A reflection across the \(y\)-axis changes the sign of the \(x\)-coordinate. Each point and its image are the same distance from the axis.
How do you rotate a figure about the origin?
- Pick the turn: \(90^{\circ}\), \(180^{\circ}\) or \(270^{\circ}\) counterclockwise.
- Apply the matching rule from the table to each vertex.
- Plot the image points and join them in the same order.
Worked example. Rotate \(A(2,1)\) by \(90^{\circ}\). The rule \((x,y)\to(-y,x)\) gives \(A^{\prime}(-1,2)\). Rotating \(B(5,1)\) gives \(B^{\prime}(-1,5)\).
| Motion | Rule for \((x,y)\) |
|---|---|
| Translation by \(a\) and \(b\) | \((x+a,\,y+b)\) |
| Reflection across the \(x\)-axis | \((x,-y)\) |
| Reflection across the \(y\)-axis | \((-x,y)\) |
| Rotation \(90^{\circ}\) | \((-y,x)\) |
| Rotation \(180^{\circ}\) | \((-x,-y)\) |
| Rotation \(270^{\circ}\) | \((y,-x)\) |
Translations, Reflections and Rotations: frequently asked questions
What stays the same after a translation, reflection or rotation?
Side lengths, angle measures and parallelism stay the same. Only the position, and for reflections and rotations the orientation, can change. That is why the image is congruent to the pre-image.
What is the rule for a reflection across the y-axis?
Change the sign of the \(x\)-coordinate: \((x,y)\to(-x,y)\). For example, \((3,-2)\) becomes \((-3,-2)\). The \(y\)-coordinate does not change.
How do you rotate a point 90 degrees counterclockwise about the origin?
Swap the coordinates and negate the new first coordinate: \((x,y)\to(-y,x)\). For example, \((4,1)\) goes to \((-1,4)\).
What is the difference between a rotation of 180 degrees and a reflection?
A \(180^{\circ}\) rotation negates both coordinates, \((-x,-y)\). A reflection across one axis negates only one coordinate. Two reflections, one across each axis, equal a \(180^{\circ}\) rotation.
What does the prime mean in A prime?
The prime marks the image. If \(A\) is a vertex of the pre-image, then \(A^{\prime}\) is the point that \(A\) moves to, and \(B^{\prime}\) matches \(B\).
How to use this grade 8 transformations lesson
Use this grade 8 transformations lesson before the worksheet, then check what you learned with the quiz.
See every chapter on the Grade 8 math page.