Transformations on the Number Plane
Translations, reflections and rotations move a shape on the number plane without changing its size. This NSW Year 9 Advanced maths subtopic describes each one with coordinates, such as a reflection in the \(x\)-axis sending \((x, y)\) to \((x, -y)\) (NESA outcome MA5‑LIN‑P‑01).
You will find the image \(P'\) of a point or triangle after a translation, a reflection in an axis or in \(y = x\), or a rotation of \(90^\circ\) or \(180^\circ\) about the origin, and after two transformations in a row. You will work backwards to an original point, name a single transformation from its effect, show that two reflections can equal a rotation, and transform the line \(y = 2x + 2\). It is part of Year 9 Advanced Properties of Geometric Figures.
Every question with a fully worked solution.
- Transformations on the Number Plane - Video - Translations Reflections and Rotations Watch
Theory
A transformation moves every point of a shape to a new position, its image. This Year 9 Advanced maths page gives the coordinate rule for each translation, reflection and rotation about the origin, then uses the rules for two transformations in a row, for working backwards, for naming a mystery transformation and for finding the image of a line. It is useful if you are heading for Advanced maths.
A transformation maps each point \(P\) to an image, written \(P'\) (read "P dash"). After a second transformation the image is \(P''\).
A translation slides every point the same distance in the same direction: right or left changes \(x\), up or down changes \(y\).
A reflection flips the plane in a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides, and the line joining them is perpendicular to it. Points on the mirror line do not move.
A rotation turns the plane about a fixed centre. Here the centre is the origin \(O\) and the angle is \(90^\circ\), \(180^\circ\) or \(270^\circ\). Anticlockwise is the positive direction.
Translations, reflections and rotations do not change lengths, angles or areas. The image is congruent to the original, so a length or an area can be worked out on whichever figure is easier.
Coordinate rules:
| Transformation | \((x, y)\) goes to |
|---|---|
| translation \(a\) right, \(b\) up | \((x + a, y + b)\) |
| reflection in the \(x\)-axis | \((x, -y)\) |
| reflection in the \(y\)-axis | \((-x, y)\) |
| reflection in \(y = x\) | \((y, x)\) |
| rotation \(90^\circ\) anticlockwise about \(O\) | \((-y, x)\) |
| rotation \(180^\circ\) about \(O\) | \((-x, -y)\) |
| rotation \(90^\circ\) clockwise about \(O\) | \((y, -x)\) |
The distance between two points, useful for comparing a point with its image:
How to transform a shape
- Write the rule for the transformation from the table, as \((x, y) \to (\ldots)\).
- Apply it to every vertex, one transformation at a time, in the order given.
- Plot the image and check it looks right: a reflection flips, a rotation turns, a translation slides.
Other question types
- Working backwards: undo the steps in reverse order. Undo a translation by moving the opposite way, a reflection by reflecting again, and a rotation by turning the same angle the other way.
- Naming a transformation: write the rule from the given points, then match it with the table. Test with a simple point such as \((1, 0)\).
- Unknown translation: apply the known transformation first, match corresponding vertices, and subtract coordinates to find how far each point moved.
- Algebraic point \((a, b)\): apply the rules with letters exactly as with numbers, then check with a numerical example.
- Lengths and areas: the image is congruent, so use the original figure if it is simpler.
Step 1. Reflect in \(y = x\): swap the coordinates, \((x, y) \to (y, x)\).
| \(A(2, 1)\) | \(\to\) | \(A'(1, 2)\) |
| \(B(5, 1)\) | \(\to\) | \(B'(1, 5)\) |
| \(C(5, 3)\) | \(\to\) | \(C'(3, 5)\) |
Step 2. Translate \(4\) units left: \((x, y) \to (x - 4, y)\).
| \(A'(1, 2)\) | \(\to\) | \(\mathbf{A''(-3, 2)}\) |
| \(B'(1, 5)\) | \(\to\) | \(\mathbf{B''(-3, 5)}\) |
| \(C'(3, 5)\) | \(\to\) | \(\mathbf{C''(-1, 5)}\) |
The order matters. Translating first and then reflecting gives a different triangle.
Look for a pattern in the coordinates. In each case the coordinates swap places and the new second coordinate changes sign.
| \((1, 4)\) | \(\to\) | \((4, -1)\) |
| \((3, 2)\) | \(\to\) | \((2, -3)\) |
| \((x, y)\) | \(\to\) | \((y, -x)\) |
Test a simple point: \((0, 1) \to (1, 0)\), a quarter-turn clockwise about \(O\).
The transformation is a rotation of \(90^\circ\) clockwise about the origin. (Anticlockwise would give \((-y, x)\), and a reflection in \(y = x\) would give \((y, x)\).)
Undo the steps in reverse order. Undo the translation first: move \(4\) left and \(2\) down.
| \(Q'\) | \(=\) | \((1 - 4,\ 5 - 2) = (-3, 3)\) |
Undo the reflection in the \(x\)-axis by reflecting again.
| \(Q\) | \(=\) | \(\mathbf{(-3, -3)}\) |
Check forwards: \((-3, -3) \to (-3, 3) \to (1, 5)\).
Reflecting in the \(x\)-axis sends \((x, y) \to (x, -y)\). A point is on the image when its reflection is on the original line, so replace \(y\) by \(-y\).
| \(-y\) | \(=\) | \(3x - 1\) |
| \(y\) | \(=\) | \(\mathbf{-3x + 1}\) |
Check with a point: \((1, 2)\) is on \(y = 3x - 1\). Its image \((1, -2)\) satisfies \(-3(1) + 1 = -2\). The gradient changes sign, and the \(x\)-intercept \(\left(\dfrac{1}{3}, 0\right)\) lies on the mirror line, so it does not move.
Common pitfalls
Frequently asked questions
What is the rule for a 90 degree rotation about the origin?
Anticlockwise, \((x, y) \to (-y, x)\). Clockwise, \((x, y) \to (y, -x)\). For example \((3, 1)\) rotated \(90^\circ\) anticlockwise is \((-1, 3)\).
How do you reflect a point in the line y = x?
Swap the coordinates: \((x, y) \to (y, x)\). The point \((5, 2)\) reflects to \((2, 5)\). Points on the line \(y = x\), such as \((4, 4)\), do not move.
What does P prime mean?
\(P'\), read "P dash" or "P prime", is the image of the point \(P\) after a transformation. After a second transformation the image is \(P''\).
Do transformations change the size of a shape?
Translations, reflections and rotations do not. The image has the same side lengths, angles and area as the original, so the two shapes are congruent.
How do you find the original point after a transformation?
Work backwards. Undo the last transformation first: move the opposite way for a translation, reflect again for a reflection, and turn the other way for a rotation. Then check by going forwards.