GCSE Transformations Practice Questions

Write down the coordinate rule for the transformation

This rule turns the words into arithmetic: apply it to each vertex in turn.

Apply the rule to vertex A

Substituting $A(1, 1)$ into the rule gives the image point $A'(4, 3)$.

State the coordinates asked for

So the image of $A$ is $\left(4,\ 3\right)$.

Write the object and the image side by side

Matching vertices are $A \rightarrow A'$, $B \rightarrow B'$ and $C \rightarrow C'$; the whole description has to be read from those three pairs.

Find the information that pins the transformation down

Subtracting the object vertex from its image gives the column vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$, and every other vertex moves by the same vector.

State the full description

Translation by the column vector $\begin{pmatrix} 4 \\ -2 \end{pmatrix}$ is the single transformation that maps triangle $ABC$ onto triangle $A'B'C'$.

Write the object and the image side by side

Matching vertices are $A \rightarrow A'$, $B \rightarrow B'$ and $C \rightarrow C'$; the whole description has to be read from those three pairs.

Decide which type of transformation it is

The shape is congruent but flipped over, so this is a reflection; the mirror line is what is left to find.

Find the information that pins the transformation down

$A$ and $A'$ have the same $x$-coordinate, and the mirror is the horizontal line halfway between them: $y = 3$.

Check the description maps every vertex correctly

The reflection in the line $y = 3$ sends all three vertices to the right places, so the description is complete and correct.

Test alternative number 1

Reflection in the line $y = 2$ would send $A(2, 1)$ to $\left(2,\ 3\right)$, but the image of $A$ is $A'(2, 5)$ — so that description is wrong.

State the full description

Reflection in the line $y = 3$ is the single transformation that maps triangle $ABC$ onto triangle $A'B'C'$.

Write the object and the image side by side

Matching vertices are $A \rightarrow A'$, $B \rightarrow B'$ and $C \rightarrow C'$; the whole description has to be read from those three pairs.

Compare a pair of matching side lengths

The image side is longer or shorter than the object side, so the shapes are not congruent — this has to be an enlargement.

Decide which type of transformation it is

The shape has changed size, so this is an enlargement; the scale factor and the centre are what is left to find.

Find the information that pins the transformation down

Matching lengths are in the ratio $2$, so $k = 2$. The lines through matching vertices all meet at $(1, 1)$.

Check the description maps every vertex correctly

The enlargement with scale factor 2, centre (1, 1) sends all three vertices to the right places, so the description is complete and correct.

Test alternative number 1

Enlargement with scale factor $2$, centre $(0, 0)$ would send $A(2, 2)$ to $\left(4,\ 4\right)$, but the image of $A$ is $A'(3, 3)$ — so that description is wrong.

Test alternative number 2

Enlargement with scale factor $3$, centre $(1, 1)$ would send $A(2, 2)$ to $\left(4,\ 4\right)$, but the image of $A$ is $A'(3, 3)$ — so that description is wrong.

Test alternative number 3

Enlargement with scale factor $-2$, centre $(1, 1)$ would send $A(2, 2)$ to $\left(-1,\ -1\right)$, but the image of $A$ is $A'(3, 3)$ — so that description is wrong.

Test alternative number 4

Enlargement with scale factor $\frac{1}{2}$, centre $(1, 1)$ would send $A(2, 2)$ to $\left(\frac{3}{2},\ \frac{3}{2}\right)$, but the image of $A$ is $A'(3, 3)$ — so that description is wrong.

State the full description

Enlargement with scale factor $2$, centre $(1, 1)$ is the single transformation that maps triangle $ABC$ onto triangle $A'B'C'$.

Identify the transformation

The question describes a enlargement with scale factor -2, centre (1, 1), but this time the IMAGE is known and the original point is not.

Write down the rule for the transformation

This rule sends the original point to the image. To go backwards it has to be undone.

Say what undoing the transformation means

Reversing a transformation means applying the transformation that takes the image back to the object.

Write down the inverse transformation

The inverse of a enlargement with scale factor -2, centre (1, 1) is a enlargement with scale factor -1/2, centre (1, 1).

Write down the rule for the inverse

This is the rule that will be applied to the image point.

Substitute the image point into the inverse rule

Putting $B'(-3, -1)$ into the inverse rule gives the original point.

Write down the original point

So the vertex that was transformed is $B(3, 2)$.

Check by applying the original transformation

Transforming $B(3, 2)$ forwards gives $(-3, -1)$, which is the image the question gave — the answer is right.

Check the distance from the key line or point

The centre, the object and the image lie on one straight line, with the distances from the centre in the ratio of the scale factor.

Identify any invariant points

The centre of enlargement $(1, 1)$ is the one point that stays fixed — every other point moves along a ray through it.

Note a second method

Instead of naming the inverse you can write the forward rule with unknowns $x$ and $y$, set it equal to the image point, and solve the two equations. The answer is the same.

Note the mistake to avoid

The scale factor multiplies the vector FROM THE CENTRE to the point, not the coordinates of the point themselves.

Sense check the position of the original point

The image is larger than the object and lies on the opposite side of the centre and upside down.

Summarise the method

To reverse a transformation, apply the inverse transformation to the image: the reverse vector, the same mirror, the opposite turn, or the reciprocal scale factor.

State the coordinates of the original point

The original point is $\left(3,\ 2\right)$.