Year 7 - Transformations on the Cartesian plane - Free practice

Year 7 Transformations on the Cartesian Plane Worksheets & Practice

Translate, reflect and rotate shapes using coordinates. Worked solutions every time, free.

Quick answer

What are transformations on the Cartesian plane?

Transformations move or flip or turn points and shapes on the Cartesian plane. The three main types are translation (sliding by a number of units horizontally and/or vertically), reflection (flipping across the x-axis or y-axis) and rotation (turning 90°, 180° or 270° about the origin). Each type changes the coordinates in a predictable way.

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Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

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Worked examples

Year 7 transformations on the cartesian plane: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Translating and reflecting a point
Point $A$ has coordinates $(3, -2)$ .

(a) Translate $A$ by 4 units left and 3 units up. Write the new coordinates.

(b) Reflect your answer from (a) across the $y$ -axis. Write the final coordinates.
Show worked solution
Key idea: A translation left subtracts from the $x$ -coordinate; a translation up adds to the $y$ -coordinate. Reflecting across the $y$ -axis changes the sign of the $x$ -coordinate only.
1. Start with $A = (3, -2)$ .
2. Translate 4 left and 3 up: $x$ -coordinate $= 3 - 4 = -1$ , $y$ -coordinate $= -2 + 3 = 1$ . So the translated point is $(-1, 1)$ .
3. Reflect $(-1, 1)$ across the $y$ -axis: change the sign of the $x$ -coordinate. $x$ -coordinate $= -(-1) = 1$ , $y$ -coordinate stays $1$ .
4. Final coordinates: $(1, 1)$ .
Answer: (a) $(-1, 1)$ (b) $(1, 1)$
Example 2 · Rotating a shape 90° about the origin
A triangle has vertices at $P(2, 1)$ , $Q(5, 1)$ and $R(5, 4)$ . Rotate the triangle $90°$ anticlockwise about the origin. Write the coordinates of the image vertices $P'$ , $Q'$ and $R'$ .
Show worked solution
Key idea: Rotating $90°$ anticlockwise about the origin maps every point $(x, y)$ to $(-y, x)$ .
1. Apply the rule $(x, y) \to (-y, x)$ to each vertex.
2. $P(2, 1) \to (-1, 2)$ , so $P' = (-1, 2)$ .
3. $Q(5, 1) \to (-1, 5)$ , so $Q' = (-1, 5)$ .
4. $R(5, 4) \to (-4, 5)$ , so $R' = (-4, 5)$ .
Answer: $P'(-1, 2)$ , $Q'(-1, 5)$ , $R'(-4, 5)$
Example 3 · Combining transformations in a real-life context
A park ranger marks the location of a water tank on a map using a coordinate grid, where each unit represents 1 kilometre. The tank is at position $(4, -3)$ .
Show worked solution
Key idea: Apply each transformation in order, one step at a time. A translation left subtracts from $x$ and a translation up adds to $y$ . Reflecting across the $x$ -axis changes the sign of the $y$ -coordinate only.
1. Start at $(4, -3)$ .
2. Translate 6 left and 2 up:
  $\begin{aligned} x &= 4 - 6 = -2 \\ y &= -3 + 2 = -1 \end{aligned}$ Position after translation: $(-2, -1)$ .
3. Reflect $(-2, -1)$ across the $x$ -axis: change the sign of the $y$ -coordinate. $x$ -coordinate stays $-2$ , $y$ -coordinate $= -(-1) = 1$ .
4. Final coordinates: $(-2, 1)$ .
Answer: $(-2, 1)$

Common mistakes

Where students usually lose marks

Mixing up which coordinate changes when reflecting

Reflecting $(3, -2)$ across the $y$ -axis gives $(3, 2)$ (changing the $y$ -coordinate instead of $x$ ).

Reflecting across the $y$ -axis only changes the sign of the $x$ -coordinate; the $y$ -coordinate stays the same. Reflecting across the $x$ -axis only changes the sign of the $y$ -coordinate. Ask yourself: which axis am I flipping over, and which coordinate measures the distance to that axis?

Getting the direction of a translation wrong

Translating $(5, 3)$ by 4 units left gives $(9, 3)$ (adding instead of subtracting from $x$ ).

Moving left means the $x$ -coordinate gets smaller, so subtract. Moving right means it gets larger, so add. The same logic applies vertically: up adds to $y$ , down subtracts from $y$ .

Confusing the 90° clockwise and anticlockwise rotation rules

Rotating $(3, 2)$ by $90°$ anticlockwise gives $(2, -3)$ , which is actually the clockwise rule.

The rule for $90°$ anticlockwise is $(x, y) \to (-y, x)$ . The rule for $90°$ clockwise is $(x, y) \to (y, -x)$ . Write the rule you need at the top of your working before you substitute any numbers.

Applying transformations in the wrong order when combining two steps

Doing the reflection before the translation when the question asks for the translation first, giving a different final answer.

Always carry out the transformations in exactly the order stated in the question. Write the result of the first transformation clearly before starting the second, so you do not lose track of which point you are transforming.

Try one yourself

Your turn

Point $P$ has coordinates $(3, -5)$ . It is reflected across the $y$ -axis and then translated $4$ units up. What is the $y$ -coordinate of the final image point?

Got a different transformations on the cartesian plane question wrong? Upload it to Tuterly and get the worked solution and a fresh one to practise, free.

About this worksheet

Why we built it

Transformations on the Cartesian plane is where Year 7 students learn how moving, flipping and turning shapes changes the coordinates of every point. These worksheets build from identifying the effect of a single transformation to describing combinations of moves, with worked solutions showing exactly how each coordinate changes so students can check their reasoning step by step.

What's covered

Sub-skills your student will practise

✓Translating points and shapes by a given number of units horizontally and vertically
✓Reflecting points and shapes across the x-axis and y-axis
✓Rotating points and shapes 90°, 180° and 270° about the origin
✓Writing the new coordinates after each transformation
✓Describing a transformation from a before-and-after set of coordinates
✓Combining two transformations and identifying the overall effect on coordinates

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Frequently asked

Questions parents ask about transformations on the cartesian plane

What are the three transformations Year 7 students need to know on the Cartesian plane?

The three main transformations are translation (sliding a shape), reflection (flipping it across an axis) and rotation (turning it about a point). At Year 7 the focus is on describing how each one changes the coordinates of every point in the shape.

How do I find the new coordinates after a reflection across the y-axis?

When you reflect a point across the y-axis, the y-coordinate stays the same and the x-coordinate changes sign. So the point (3, 5) becomes (−3, 5). Reflecting across the x-axis works the same way but the x-coordinate stays and the y-coordinate changes sign.

What does rotating a point 90° about the origin do to its coordinates?

A 90° anticlockwise rotation turns the point (x, y) into (−y, x). A 90° clockwise rotation gives (y, −x). Practising these rules with specific coordinates is one of the most common worksheet activities at this level.

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