NZ Level 5

Transformations on the Cartesian Plane

Lesson

A transformation is a change, so when we transform a shape, we change it in some way. There are three kinds of transformations: reflections, rotations and translations. Let's recap these first.

We see reflections all the time- in mirrors, in pools of water and so on. A flip is a reflection over a line or axis. We can see in the picture below that the blue object has been reflected over the vertical axis to create the green image. Notice how they are exactly the same distance from the $y$`y`-axis?

A shape is rotated around a centre point in a circular motion. It does not have to be turned in a full circle, otherwise it would be back and the same point. We commonly see $90^\circ$90° turns (also known as quarter turns), $180^\circ$180° turns (half turns) and $270^\circ$270° turns (three-quarter turns). As we can see the blue object has been rotated around the origin, $\left(0,0\right)$(0,0).

The whole shape moves the same distance in the same direction, without being rotated or flipped. In the picture below, we can see the object has been moved up $5$5 units.

After any of those transformations (rotations, reflections and translations), the shape still has the same size, area, angles and line lengths. However, a shape may be transformed in more than one way.

Here are some more worked examples.

What is the translation of the trapezium ABCD to the trapezium EFGH?

Plot the new triangle formed by reflecting the given triangle about the line $x=1$`x`=1.

a) Graph the points A$\left(1,5\right)$(1,5), B$\left(9,5\right)$(9,5), C$\left(9,9\right)$(9,9) and D$\left(5,9\right)$(5,9).

b) Then plot the points A', B', C'and D" that would result when we rotate the original points A,B,C and D, $90^\circ$90° clockwise.

a) Graph the points A$\left(-5,-5\right)$(−5,−5), B$\left(-1,-5\right)$(−1,−5), C$\left(-1,-1\right)$(−1,−1) and D$\left(-5,-1\right)$(−5,−1).

b)Then plot the points A'B'C'and D" that would result when we rotate the original points A,B,C and D, $90^\circ$90°clockwise.

Define and use transformations and describe the invariant properties of figures and objects under these transformations