Transformations

Object, image and the four transformations

A transformation moves or changes a shape. The starting shape is the object and the result is the image. In this topic there are four transformations.

• Reflection flips the shape over a mirror line.
• Rotation turns the shape about a fixed point, the centre of rotation.
• Translation slides the shape without turning it.
• Enlargement changes the size of the shape from a fixed point, the centre of enlargement.

Reflections, rotations and translations keep every length and every angle, so the image is congruent to the object: identical in shape and size. An enlargement keeps the angles but multiplies every length by the scale factor, so the image is similar to the object.

A quick way to tell them apart when you see an object and its image:

1. Different size? It is an enlargement.
2. Same size, and every side points the same way as before? It is a translation.
3. Same size but a mirror image (the vertices go round in the opposite order)? It is a reflection.
4. Same size, turned, but not a mirror image? It is a rotation.

Always use a ruler for straight edges when you draw an image, and label it as the question asks.

Reflection

In a reflection every point of the object goes to the same perpendicular distance on the other side of the mirror line. A point on the mirror line stays where it is.

To reflect in a vertical line $x=k$, only the $x$-coordinate changes: count how far each vertex is from the line and go the same distance on the other side. To reflect in a horizontal line $y=k$, only the $y$-coordinate changes. Remember that $x=k$ is vertical and $y=k$ is horizontal; the $y$-axis is $x=0$ and the $x$-axis is $y=0$.

Why it works: the mirror line is the perpendicular bisector of the join of each point and its image, so it is exactly halfway between them. That also tells you how to find a mirror line: take any vertex and its image, and find the midpoint. For example, if $(5,2)$ reflects to $(-1,2)$, the mirror line is $x=\frac{5+(-1)}{2}=2$.

To describe a reflection fully you need two things: the word reflection and the equation of the mirror line.

Rotation

A rotation turns every point through the same angle about a fixed centre. You need three things to describe one fully: the word rotation, the angle with its direction (clockwise or anticlockwise), and the centre. A half turn of ${180}^{°}$ is the same in both directions, so it needs no direction.

The reliable way to draw a rotation is with tracing paper: trace the shape, hold the tracing at the centre with your pencil point, turn it through the angle, and mark where the vertices land. Without tracing paper, work with the vector from the centre to each vertex. A quarter turn anticlockwise sends $\left(\begin{array}{c}a\\ b\end{array}\right)$ to $\left(\begin{array}{c}-b\\ a\end{array}\right)$, and a quarter turn clockwise sends $\left(\begin{array}{c}a\\ b\end{array}\right)$ to $\left(\begin{array}{c}b\\ -a\end{array}\right)$. A half turn sends $\left(\begin{array}{c}a\\ b\end{array}\right)$ to $\left(\begin{array}{c}-a\\ -b\end{array}\right)$.

When the centre is the origin, those become rules for the coordinates:

• ${90}^{°}$ anticlockwise: $(x,y)\to (-y,x)$
• ${90}^{°}$ clockwise: $(x,y)\to (y,-x)$
• ${180}^{°}$: $(x,y)\to (-x,-y)$

At Core level the centre is the origin, a vertex of the shape, or the midpoint of one of its edges. If the centre is a vertex, that vertex does not move. For a half turn, the centre is the midpoint of every point and its image.

Translation

A translation slides every point the same distance in the same direction. It is described by a column vector $\left(\begin{array}{c}x\\ y\end{array}\right)$: the top number is the movement to the right (negative means left) and the bottom number is the movement up (negative means down). For example, $\left(\begin{array}{c}-4\\ 3\end{array}\right)$ means 4 to the left and 3 up.

To find the vector, pick a vertex of the object and the matching vertex of the image, and subtract: image coordinates minus object coordinates. Check with a second vertex. The vector from the image back to the object has both signs changed.

To describe a translation fully: the word translation and the column vector. Do not write it as coordinates: $(3,-5)$ is a point, $\left(\begin{array}{c}3\\ -5\end{array}\right)$ is a movement.

Enlargement

An enlargement multiplies the distance of every point from the centre of enlargement by the scale factor. The image is the same shape as the object, and every length is multiplied by the scale factor.

• Scale factor greater than 1: the image is bigger and further from the centre.
• Scale factor between 0 and 1, such as $\frac{1}{2}$: the image is smaller and closer to the centre. It is still called an enlargement.

To draw an enlargement with centre C and scale factor $k$: for each vertex, find the vector from C to the vertex, multiply it by $k$, and add the result to C. For example, with centre $(1,2)$ and scale factor 3, the vertex $(3,3)$ is $\left(\begin{array}{c}2\\ 1\end{array}\right)$ from the centre, so its image is $3\times \left(\begin{array}{c}2\\ 1\end{array}\right)=\left(\begin{array}{c}6\\ 3\end{array}\right)$ from the centre, at $(7,5)$.

To find the scale factor, divide a length on the image by the matching length on the object.

To find the centre, draw straight lines through each vertex and its image and extend them. They all meet at the centre. This works because each image point lies on the ray from the centre through the object point.

To describe an enlargement fully: the word enlargement, the scale factor and the centre.

Describing a transformation fully

Questions that say describe fully the single transformation give a mark for the name and a mark for each piece of information. Learn the list:

• Reflection: the equation of the mirror line.
• Rotation: the angle, the direction (unless it is ${180}^{°}$) and the centre.
• Translation: the column vector.
• Enlargement: the scale factor and the centre.

Name only one transformation. An answer such as "a rotation and then a translation" scores nothing when one transformation is asked for, even if the combination would work. Give the centre as coordinates and the mirror line as an equation, never as a description such as "the dotted line".