Year 7 Maths Number Plane

Transformations

Q: What happens to the coordinates in a reflection in the x-axis?

The x-coordinate stays and the y-coordinate changes sign: (x, y) → (x, -y).

Q: What happens to the coordinates in a reflection in the y-axis?

The y-coordinate stays and the x-coordinate changes sign: (x, y) → (-x, y).

Q: What is a rotation of 180 degrees about the origin?

A half turn. Both coordinates change sign: (x, y) → (-x, -y).

20 practice questions 2 video lessons Theory + worked examples

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Transformations - Reflection Watch
Transformations - Translating A Triangle On The Coordinate Plane Watch

Theory

A transformation moves a shape to a new position. The new shape is the image, and image points get a dash: \(A\) moves to \(A^{\prime}\) (“\(A\) dash”). This Year 7 maths page shows how to translate (slide), reflect (flip) and rotate (turn) shapes on the number plane using coordinate rules.

Translation (slide). Every point moves the same distance in the same direction. Right and left change \(x\). Up and down change \(y\).

Reflection (flip) in an axis (also said “about” or “across” an axis). Each point lands the same distance from the mirror line, on the other side. In the \(x\)-axis, \(x\) stays and \(y\) changes sign. In the \(y\)-axis, \(y\) stays and \(x\) changes sign.

Rotation (turn). Every point turns through the same angle about a centre of rotation. About the origin, a half turn (\(180^\circ\)) changes both signs. For a turn about another point, count each vertex’s moves from the centre, then turn those moves.

Apply the rule to every vertex, then join the image points. The image is congruent to the original: the same side lengths and angles.

Triangle T (solid) and three images (dashed): reflected in the \(y\)-axis (top left), reflected in the \(x\)-axis (bottom right) and rotated \(180^\circ\) about \(O\) (bottom left). Each image is the same size and shape as T.

Where the point \((4, 1)\) goes:

Transformation	Rule	\((4, 1)\) goes to
Translate \(3\) right, \(1\) down	\((x + 3,\ y - 1)\)	\((7, 0)\)
Reflect in the \(x\)-axis	\((x,\ -y)\)	\((4, -1)\)
Reflect in the \(y\)-axis	\((-x,\ y)\)	\((-4, 1)\)
Rotate \(180^\circ\) about \(O\)	\((-x,\ -y)\)	\((-4, -1)\)
Rotate \(90^\circ\) clockwise about \(O\)	\((y,\ -x)\)	\((1, -4)\)

          Naming a transformation. Compare the coordinates before and after. The same change to every point: translation. One sign changes: reflection. Both signs change: rotation of \(180^\circ\) about the origin.
        

          Two reflections make a half turn. Reflect in the \(x\)-axis, then in the \(y\)-axis: both signs change, so the result is the same as a rotation of \(180^\circ\) about the origin.
        

          Undoing a translation. Move the same distances the opposite way: the reverse of “\(2\) right, \(5\) up” is “\(2\) left, \(5\) down”.
        

How to transform a shape

Write the rule, for example \((x, y) \to (x + 5,\ y - 3)\).
Apply it to every vertex and label the images \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\).
Plot and join the image points in the same order.
Check one side length: the image is congruent.

Example 1 — Translation

Triangle \(ABC\) has \(A(-4, 1)\), \(B(-1, 1)\) and \(C(-4, 3)\). It is translated \(5\) units right and \(3\) units down. Find \(A^{\prime}\), \(B^{\prime}\) and \(C^{\prime}\).

Solution

Right changes \(x\); down changes \(y\). The rule is \((x, y) \to (x + 5,\ y - 3)\).

\(A^{\prime}(1, -2)\), \(B^{\prime}(4, -2)\), \(C^{\prime}(1, 0)\). Check: each \(x\) is \(5\) more and each \(y\) is \(3\) less.

Example 2 — Reflection in an axis

Triangle \(PQR\) has \(P(1, 1)\), \(Q(4, 2)\) and \(R(2, 4)\).
(i) Reflect it in the \(x\)-axis.
(ii) Where would \(R\) go in a reflection in the \(y\)-axis?

Solution

(i) \(x\) stays, \(y\) changes sign: \((x, y) \to (x,\ -y)\).
(ii) In the \(y\)-axis, \(y\) stays and \(x\) changes sign.

(i) \(P^{\prime}(1, -1)\), \(Q^{\prime}(4, -2)\), \(R^{\prime}(2, -4)\). \(R\) is \(4\) above the axis and \(R^{\prime}\) is \(4\) below.
(ii) \(R(2, 4) \to (-2, 4)\)

Example 3 — Quarter turn about the origin

Triangle \(ABC\) has \(A(2, 1)\), \(B(5, 1)\) and \(C(2, 3)\). It is rotated \(90^\circ\) clockwise about the origin \(O\). Find the image.

Solution

Use the rule for \(90^\circ\) clockwise: \((x, y) \to (y,\ -x)\).

\(A^{\prime}(1, -2)\), \(B^{\prime}(1, -5)\), \(C^{\prime}(3, -2)\). Check: \(AB\) went across \(5 - 2 = 3\); the image side \(A^{\prime} B^{\prime}\) goes down \(-2 - (-5) = 3\).

Example 4 — Half turn about a point

Triangle \(ABC\) has \(A(3, 2)\), \(B(5, 2)\) and \(C(3, 4)\). It is rotated \(180^\circ\) about \(M(4, 1)\).
(i) Find \(A^{\prime}\), \(B^{\prime}\) and \(C^{\prime}\).
(ii) Is \(A^{\prime} B^{\prime} C^{\prime}\) congruent to \(ABC\)?

Solution

(i) Count each move from \(M\), then reverse it. \(A\) is \(1\) left and \(1\) up from \(M\), so \(A^{\prime}\) is \(1\) right and \(1\) down from \(M\).

\(A'\)	\(=\)	\((4 + 1,\ 1 - 1) = (5,\ 0)\)
\(B'\)	\(=\)	\((4 - 1,\ 1 - 1) = (3,\ 0)\)
\(C'\)	\(=\)	\((4 + 1,\ 1 - 3) = (5,\ -2)\)

A' = (5, 0), B' = (3, 0), C' = (5, - 2)

(i) \(A^{\prime}(5, 0)\), \(B^{\prime}(3, 0)\), \(C^{\prime}(5, -2)\)
(ii) Yes. A rotation only turns the shape: \(AB = 2\) and \(A^{\prime} B^{\prime} = 5 - 3 = 2\).

Example 5 — Name the transformation

The table shows two points before and after a transformation: \((3, 2) \to (3, -2)\) and \((5, 4) \to (5, -4)\).
(i) Which transformation was used?
(ii) Where does \((-1, 7)\) go?
(iii) Name a point that is its own image.

Solution

\(x\) stays the same and \(y\) changes sign, so the mirror is the \(x\)-axis. Points on the mirror line do not move.

(i) A reflection in the \(x\)-axis
(ii) \((-1, -7)\)
(iii) Any point on the \(x\)-axis, for example \((4, 0)\)

Example 6 — Two reflections

The point \(D(-2, 5)\) is reflected in the \(y\)-axis to \(D^{\prime}\), then \(D^{\prime}\) is reflected in the \(x\)-axis to \(D^{\prime\prime}\). Find \(D^{\prime}\) and \(D^{\prime\prime}\). Which single transformation moves \(D\) to \(D^{\prime\prime}\)?

Solution

In the \(y\)-axis, \(x\) changes sign: \(D^{\prime}(2, 5)\). In the \(x\)-axis, \(y\) changes sign: \(D^{\prime\prime}(2, -5)\).

Both signs have changed, \((-2, 5) \to (2, -5)\): a rotation of \(180^\circ\) about the origin.

Common pitfalls

Reflecting in the x-axis changes y, not x. The mirror is the \(x\)-axis, so points flip up or down: \((4, 1) \to (4, -1)\).

Mixing up the directions. “\(4\) units down” means \(y - 4\). “\(4\) units left” means \(x - 4\).

Turning the wrong way. Check with one point. In a quarter turn clockwise about \(O\), a point to the right of \(O\) moves to below \(O\): \((3, 0) \to (0, -3)\).

Moving only one vertex. Apply the rule to every vertex, then join the image points in the same order.

Frequently asked questions

What is a translation?

A slide: every point moves the same distance in the same direction.

What happens to the coordinates in a reflection in the x-axis?

The \(x\)-coordinate stays and the \(y\)-coordinate changes sign: \((x, y) \to (x, -y)\).

What happens to the coordinates in a reflection in the y-axis?

The \(y\)-coordinate stays and the \(x\)-coordinate changes sign: \((x, y) \to (-x, y)\).

What is a rotation of 180 degrees about the origin?

A half turn. Both coordinates change sign: \((x, y) \to (-x, -y)\).

Is the image congruent to the original?

Yes. Translations, reflections and rotations keep the same side lengths and angles.

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Transformations

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