Year 7AC9M7SP03

Transformations on the Cartesian Plane

Describe transformations of a set of points using coordinates in the Cartesian plane, translations and reflections on an axis, and rotations about a given point.

A transformation moves a shape without changing its size. Describe it by what happens to the coordinates and you can never get it wrong.

Method

Translation slides

(x,y) \rightarrow (x+a,\ y+b)

Add to the coordinates. Nothing turns or flips.

Reflection in the $x$-axis

(x,y) \rightarrow (x,\ -y)

The $y$ coordinate changes sign.

Reflection in the $y$-axis

(x,y) \rightarrow (-x,\ y)

The $x$ coordinate changes sign.

Rotation of $180^\circ$ about the origin

(x,y) \rightarrow (-x,\ -y)

Both signs change.

Worked example

e.g.

\text{Reflect } A(3,2) \text{ in the } x\text{-axis, then translate it } 4 \text{ left and } 1 \text{ up.}

Write the starting point. $A(3,2)$
Reflect in the $x$ -axis: change the sign of $y$ . $(3,-2)$
Translate 4 left: subtract 4 from $x$ . $(-1,-2)$
Translate 1 up: add 1 to $y$ . $(-1,-1)$
State the final image. $A''(-1,-1)$

Practice

Apply each rule to the coordinates.

\text{Reflect } (5,3) \text{ in the } y\text{-axis}

Answer

(-5,3)

\text{Translate } (2,-4) \text{ by } 3 \text{ right, } 5 \text{ up}

Answer

(5,1)

\text{Rotate } (4,1) \text{ by } 180^\circ \text{ about the origin}

Answer

(-4,-1)

\text{Which transformation maps } (2,7) \text{ to } (2,-7)?

Answer

\text{reflection in the } x\text{-axis}

\text{Does a translation change a shape's size?}

Answer

\text{No — only its position.}