Year 12 Coordinate Geometry: Image of a point

Image questions describe what happens to a point under a geometric transformation. The transformation might be a translation, reflection, rotation, enlargement, or a sequence of two or more of these. The most reliable strategy is to apply each transformation one at a time and write the intermediate point clearly. Trying to combine everything mentally often leads to sign errors.

Translate by adding a vector. Reflect by changing coordinates according to the mirror line. Rotate by using a known coordinate rule when the centre is the origin.

Useful coordinate rules

Reflection in the x-axis: $(x, y)$ becomes $(x, - y)$ .
Reflection in the y-axis: $(x, y)$ becomes $(- x, y)$ .
Reflection in the line $y = x$ : $(x, y)$ becomes $(y, x)$ .
Translation by $(a, b)$ : $(x, y)$ becomes $(x + a, y + b)$ .
Rotation 180 degrees about the origin: $(x, y)$ becomes $(- x, - y)$ .

Worked example 1: reflection then translation

Problem: Point

P (3, - 1)

is reflected in the y-axis and then translated by the vector

(- 2, 4)

. Find the image point

P'

Reflection in the y-axis changes $(x, y)$ to $(- x, y)$ .
So the reflected point is $(- 3, - 1)$ .
Now translate by $(- 2, 4)$ : $P' = (- 3, - 1) + (- 2, 4)$ .
This gives $P' = (- 5, 3)$ .
Check the logic. The reflection moves the point to the left side of the axis, and the translation moves it further left and upward, so the answer is sensible.

Answer:

P' = (- 5, 3)

Worked example 2: swapping coordinates

Problem: Point

Q (- 4, 7)

is reflected in the line

y = x

. Then it is translated by

(3, - 2)

. Find the final image.

Reflection in $y = x$ swaps the coordinates, so $Q$ becomes $(7, - 4)$ .
Translate by $(3, - 2)$ : $(7, - 4) + (3, - 2) = (10, - 6)$ .
The final image is $(10, - 6)$ .

Answer:

(10, - 6)

Why order matters

Transformations do not always commute. If you translate first and then reflect, you may not get the same answer as reflecting first and then translating. That is why the wording of the question matters. Follow the order exactly as given. When several steps are involved, label the intermediate images as $P 1$ , $P 2$ and so on.

Common mistakes

Changing both signs when reflecting in the y-axis instead of just the x-coordinate.
Forgetting that translation means add the vector, not multiply by it.
Applying the correct rules in the wrong order.
Confusing reflection in $y = x$ with reflection in the x-axis.

Exam habit

Sketching a small set of axes can help immediately. Even a rough picture tells you whether a point should end up above or below an axis, left or right of the origin, or closer to a mirror line. That visual check is useful when your algebra produces a sign you were not expecting.

This topic gets easier when every transformation is treated as a precise rule, not as a visual guess. Once you know the coordinate effect of a reflection, rotation, or translation, multi-step image questions become bookkeeping rather than intuition.

Revision focus

A strong Year 12 habit is to move between representations instead of trusting just one. In algebra, that means checking whether each step preserves the same meaning. In functions and trigonometry, it means linking equations, graphs, intervals, and diagrams. In coordinate geometry, it means asking whether the computed coordinates still match the geometric story in the question. When several representations agree, your answer is usually on solid ground.

It also helps to add one final sentence of interpretation after the working is done. State what the solution means in the language of the problem, not just in symbols. That last step slows you down just enough to catch sign errors, missing restrictions, interval mistakes, or values that are mathematically correct but contextually impossible.

Practice links

Image of a point