Transformations · Vectors & transformations

A transformation moves every point of a shape by the same rule. The shape you start with is the object and the result is the image. WAEC questions usually give the vertices, ask you to draw the image, and then to describe a transformation.

Translation

A translation slides the shape without turning it. A translation by $\begin{pmatrix} a \\ b \end{pmatrix}$ adds $a$ to every $x$ and $b$ to every $y$ : $(x, y) \to (x + a, y + b)$ .

TranslationEvery point moves by the same vector

Reflection

A reflection flips the shape over a mirror line. Each image point is the same distance from the line as the object point, on the other side. The four mirror lines WAEC uses:

Mirror line	Rule
the $x$ -axis	$(x, y) \to (x, -y)$
the $y$ -axis	$(x, y) \to (-x, y)$
$y = x$	$(x, y) \to (y, x)$
$y = -x$	$(x, y) \to (-y, -x)$

Reflection in y = xThe coordinates swap: (3, 1) goes to (1, 3)

Rotation

A rotation turns the shape about a centre. About the origin:

Rotation about $O$	Rule
$90^\circ$ anticlockwise	$(x, y) \to (-y, x)$
$90^\circ$ clockwise	$(x, y) \to (y, -x)$
$180^\circ$	$(x, y) \to (-x, -y)$

Rotation 90° anticlockwise(3, 1) goes to (−1, 3): swap, then change the sign of the new x

Enlargement

An enlargement from the origin with scale factor $k$ multiplies every coordinate by $k$ : $(x, y) \to (kx, ky)$ . A scale factor between $-1$ and $1$ makes the shape smaller. A negative scale factor puts the image on the other side of the centre, upside down.

Enlargement from OScale factor 2 (teal) and −½ (orange): lines from O pass through matching points

Try it

TransformationsPick a transformation

(x, y) → (y, x)ruleA′(1, 1) B′(1, 4) C′(3, 1)image

A reflection in the line y = x: each image point is the same distance from the mirror, on the other side. The rule is (x, y) → (y, x). The image is flipped over.

Try each transformation and read its rule. Compare “Rotate 180°” with “Enlarge” at scale factor $-1$ : they give the same image.

Combining and describing

To combine two transformations, apply the first rule, then apply the second rule to the result. To describe a single transformation, give its type and every detail: the vector of a translation, the mirror line of a reflection, the centre, angle and direction of a rotation, or the centre and scale factor of an enlargement.

Worked example · WAEC 2016

WAEC 2016 · Paper 2 · Q13 (b)

Draw on this graph, indicating the coordinates of all vertices: (i) the quadrilateral $PQRS$ with vertices $P(-5, -4)$ , $Q(2, -1)$ , $R(0, 3)$ and $S(-8, 4)$ ; (ii) the image $P_1Q_1R_1S_1$ of $PQRS$ under a translation by the vector $\begin{pmatrix} 3 \\ -8 \end{pmatrix}$ ; (iii) the image $P_2Q_2R_2S_2$ of $PQRS$ under an enlargement from the origin with scale factor $-\frac12$ .

(ii) The translation
$P_1(-2, -12)$ , $Q_1(5, -9)$ , $R_1(3, -5)$ , $S_1(-5, -4)$ .
Think first. Add 3 to each x and take 8 from each y.
(iii) The enlargement
$P_2(2.5, 2)$ , $Q_2(-1, 0.5)$ , $R_2(0, -1.5)$ , $S_2(4, -2)$ : on the other side of the origin, half the size.
Think first. Multiply each coordinate of PQRS by −½.

Worked example · WAEC 2018

WAEC 2018 · Paper 2 · Q13 (b, c)

Given the points $P(3, 2)$ , $Q(-1, 5)$ , $R(0, 8)$ and $S(3, 7)$ , draw on the same graph, indicating clearly the vertices and their coordinates, the: (i) quadrilateral $PQRS$ ; (ii) image $P_1Q_1R_1S_1$ of $PQRS$ under an anticlockwise rotation of $90^\circ$ about the origin; (iii) image $P_2Q_2R_2S_2$ of $P_1Q_1R_1S_1$ under a reflection in the line $y - x = 0$ .

Describe precisely the single transformation $T$ for which $T : PQRS \to P_2Q_2R_2S_2$ .

(ii) Rotate 90° anticlockwise
$P_1(-2, 3)$ , $Q_1(-5, -1)$ , $R_1(-8, 0)$ , $S_1(-7, 3)$ .
Think first. Use (x, y) → (−y, x).
(iii) Reflect in y = x
$P_2(3, -2)$ , $Q_2(-1, -5)$ , $R_2(0, -8)$ , $S_2(3, -7)$ .
Think first. y − x = 0 is the line y = x: swap the coordinates.
(c) The single transformation
Each point $(x, y)$ has gone to $(x, -y)$ : $T$ is a reflection in the $x$ -axis.
Think first. Compare P(3, 2) with P₂(3, −2).

Your turn

WAEC 2018 · Paper 2 · Q12 (b)

Vectors & transformations

(b)
Using a scale of 2 cm to 2 units on both axes, draw on a graph paper two perpendicular axes $Ox$ and $Oy$ for $-8 \le x \le 8$ and $-8 \le y \le 8$ respectively. Draw, on the same graph paper, indicating clearly the vertices and their coordinates: (i) the quadrilateral $WXYZ$ with $W(2, 3)$ , $X(4, -1)$ , $Y(-3, -4)$ and $Z(-3, 2)$ ; (ii) the image $W_1X_1Y_1Z_1$ of the quadrilateral $WXYZ$ under an anticlockwise rotation of $90^\circ$ about the origin.
Model answer
A $90^\circ$ anticlockwise rotation about the origin maps $(x, y)$ to $(-y, x)$ . The image (dashed) is $W_1(-3, 2)$ , $X_1(1, 4)$ , $Y_1(4, -3)$ , $Z_1(-2, -3)$ ; note that $W_1$ lands on the same point as $Z$ .

Try it on a graph

WXYZ (blue) and its image under a 90° anticlockwise rotation about O (red).

Worked solution (try it first)

(b)

With 2 cm to 2 units, draw both axes from $-8$ to $8$ .

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