Transformations
Transformations moves points and shapes around the number plane. In NSW Year 7 maths it consolidates primary-school work on slides, flips and turns, now described exactly with coordinates. A translation slides a shape, a reflection flips it in an axis and a rotation turns it about a point.
You will translate points and shapes, reflect them in the \(x\)-axis and the \(y\)-axis, rotate them \(180^\circ\) or \(90^\circ\) about the origin or another point, describe each move with a coordinate rule such as \((x, y) \to (x, -y)\), and see that the image has the same size and shape as the original. Tile patterns and logos use these moves. It is part of Year 7 Number Plane.
Theory
A transformation moves a shape to a new position. The new shape is the image, and image points get a dash: \(A\) moves to \(A^{\prime}\) (“\(A\) dash”). This Year 7 maths page shows how to translate (slide), reflect (flip) and rotate (turn) shapes on the number plane using coordinate rules.
Translation (slide). Every point moves the same distance in the same direction. Right and left change \(x\). Up and down change \(y\).
Reflection (flip) in an axis (also said “about” or “across” an axis). Each point lands the same distance from the mirror line, on the other side. In the \(x\)-axis, \(x\) stays and \(y\) changes sign. In the \(y\)-axis, \(y\) stays and \(x\) changes sign.
Rotation (turn). Every point turns through the same angle about a centre of rotation. About the origin, a half turn (\(180^\circ\)) changes both signs. For a turn about another point, count each vertex’s moves from the centre, then turn those moves.
Apply the rule to every vertex, then join the image points. The image is congruent to the original: the same side lengths and angles.
Where the point \((4, 1)\) goes:
| Transformation | Rule | \((4, 1)\) goes to |
|---|---|---|
| Translate \(3\) right, \(1\) down | \((x + 3,\ y - 1)\) | \((7, 0)\) |
| Reflect in the \(x\)-axis | \((x,\ -y)\) | \((4, -1)\) |
| Reflect in the \(y\)-axis | \((-x,\ y)\) | \((-4, 1)\) |
| Rotate \(180^\circ\) about \(O\) | \((-x,\ -y)\) | \((-4, -1)\) |
| Rotate \(90^\circ\) clockwise about \(O\) | \((y,\ -x)\) | \((1, -4)\) |
How to transform a shape
- Write the rule, for example \((x, y) \to (x + 5,\ y - 3)\).
- Apply it to every vertex and label the images \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\).
- Plot and join the image points in the same order.
- Check one side length: the image is congruent.
Right changes \(x\); down changes \(y\). The rule is \((x, y) \to (x + 5,\ y - 3)\).
\(A^{\prime}(1, -2)\), \(B^{\prime}(4, -2)\), \(C^{\prime}(1, 0)\). Check: each \(x\) is \(5\) more and each \(y\) is \(3\) less.
(i) Reflect it in the \(x\)-axis.
(ii) Where would \(R\) go in a reflection in the \(y\)-axis?
(i) \(x\) stays, \(y\) changes sign: \((x, y) \to (x,\ -y)\).
(ii) In the \(y\)-axis, \(y\) stays and \(x\) changes sign.
(i) \(P^{\prime}(1, -1)\), \(Q^{\prime}(4, -2)\), \(R^{\prime}(2, -4)\). \(R\) is \(4\) above the axis and \(R^{\prime}\) is \(4\) below.
(ii) \(R(2, 4) \to (-2, 4)\)
Use the rule for \(90^\circ\) clockwise: \((x, y) \to (y,\ -x)\).
\(A^{\prime}(1, -2)\), \(B^{\prime}(1, -5)\), \(C^{\prime}(3, -2)\). Check: \(AB\) went across \(5 - 2 = 3\); the image side \(A^{\prime} B^{\prime}\) goes down \(-2 - (-5) = 3\).
(i) Find \(A^{\prime}\), \(B^{\prime}\) and \(C^{\prime}\).
(ii) Is \(A^{\prime} B^{\prime} C^{\prime}\) congruent to \(ABC\)?
(i) Count each move from \(M\), then reverse it. \(A\) is \(1\) left and \(1\) up from \(M\), so \(A^{\prime}\) is \(1\) right and \(1\) down from \(M\).
| \(A'\) | \(=\) | \((4 + 1,\ 1 - 1) = (5,\ 0)\) |
| \(B'\) | \(=\) | \((4 - 1,\ 1 - 1) = (3,\ 0)\) |
| \(C'\) | \(=\) | \((4 + 1,\ 1 - 3) = (5,\ -2)\) |
(i) \(A^{\prime}(5, 0)\), \(B^{\prime}(3, 0)\), \(C^{\prime}(5, -2)\)
(ii) Yes. A rotation only turns the shape: \(AB = 2\) and \(A^{\prime} B^{\prime} = 5 - 3 = 2\).
(i) Which transformation was used?
(ii) Where does \((-1, 7)\) go?
(iii) Name a point that is its own image.
\(x\) stays the same and \(y\) changes sign, so the mirror is the \(x\)-axis. Points on the mirror line do not move.
(i) A reflection in the \(x\)-axis
(ii) \((-1, -7)\)
(iii) Any point on the \(x\)-axis, for example \((4, 0)\)
In the \(y\)-axis, \(x\) changes sign: \(D^{\prime}(2, 5)\). In the \(x\)-axis, \(y\) changes sign: \(D^{\prime\prime}(2, -5)\).
Both signs have changed, \((-2, 5) \to (2, -5)\): a rotation of \(180^\circ\) about the origin.
Common pitfalls
Frequently asked questions
What is a translation?
A slide: every point moves the same distance in the same direction.
What happens to the coordinates in a reflection in the x-axis?
The \(x\)-coordinate stays and the \(y\)-coordinate changes sign: \((x, y) \to (x, -y)\).
What happens to the coordinates in a reflection in the y-axis?
The \(y\)-coordinate stays and the \(x\)-coordinate changes sign: \((x, y) \to (-x, y)\).
What is a rotation of 180 degrees about the origin?
A half turn. Both coordinates change sign: \((x, y) \to (-x, -y)\).
Is the image congruent to the original?
Yes. Translations, reflections and rotations keep the same side lengths and angles.