TransformationsIB MYP Maths Extended: Revision notes

A transformation moves or changes a shape. The original is the object and the new shape is the image. We write $A'$ for the image of $A$. Translations, reflections and rotations keep the shape the same size (the image is congruent to the object). An enlargement changes the size but keeps the shape.

A translation slides every point the same distance in the same direction. Describe it with a movement, for example '$3$ right and $2$ up'. To translate a point, add the movement to its coordinates: $(1,4)$ moved $3$ right and $2$ down becomes $(4,2)$. To describe a translation, compare one point with its image. Right and up are positive, left and down are negative.

A reflection flips a shape in a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. Useful rules:

• In the $x$-axis: $(x,y)\to(x,-y)$.
• In the $y$-axis: $(x,y)\to(-x,y)$.
• In the line $y=x$: $(x,y)\to(y,x)$.
• In a vertical line $x=a$: the $y$-coordinate stays the same and the new $x$-coordinate is $2a-x$. For example $(1,1)$ in $x=5$ gives $(9,1)$. To describe a reflection, give the equation of the mirror line, which is halfway between a point and its image.

A rotation turns a shape about a fixed point called the centre of rotation. To describe it you need three things: the angle, the direction (clockwise or anticlockwise) and the centre. About the origin: $90^\circ$ anticlockwise takes $(x,y)$ to $(-y,x)$, $90^\circ$ clockwise takes $(x,y)$ to $(y,-x)$ and $180^\circ$ takes $(x,y)$ to $(-x,-y)$. For other centres, count the position of the point relative to the centre, then turn that movement. A $180^\circ$ turn about $(1,1)$ sends $(3,2)$, which is $2$ right and $1$ up from the centre, to $2$ left and $1$ down: $(-1,0)$.

An enlargement changes the size of a shape. You need a scale factor and a centre of enlargement. Every distance from the centre is multiplied by the scale factor, so lengths in the image are the scale factor times the lengths in the object. At this level the scale factor is a positive whole number. If the centre is the origin, multiply each coordinate by the scale factor: scale factor $3$ takes $(3,2)$ to $(9,6)$. For another centre, multiply the distance from the centre. With centre $(1,1)$ and scale factor $2$, the point $(3,2)$ is $2$ right and $1$ up from the centre, so its image is $4$ right and $2$ up: $(5,3)$. A point at the centre does not move. The image is similar to the object.

To describe a single transformation fully, give all the details: translation (the movement), reflection (the mirror line), rotation (angle, direction, centre) and enlargement (scale factor, centre). To find a centre of rotation, join a point to its image, draw the perpendicular bisector of that line, do the same for a second point, and the centre is where the two bisectors cross. For a $180^\circ$ rotation, the centre is the midpoint of any point and its image: $(1,2)\to(5,4)$ has centre $(3,3)$.