NZ Level 5

Combined 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 transformations. 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. Note that a shape may be transformed in more than one way.

Below are some worked examples where there is more than one transformation.

For the point on the plane, plot the point that will result from a translation of $4$4 units right and $2$2 units up, followed by a rotation of $90^\circ$90° anticlockwise.

Plot the triangle that will result when the original triangle is reflected on the x-axis and translated 1 unit right.

Triangle ABC is first reflected about the given line to form triangle DEF.

a) Plot the triangle DEF

b) Triangle DEF is reflected about the y-axis to form triangle GHI. What single transformation would overlap triangle ABC onto triangle GHI?

Point A$\left(-9,7\right)$(−9,7) is rotated $90^\circ$90° clockwise about the origin to form point B.

a) What are the coordinates of point B?

b) Point B is then translated to the left so that it lies on the same vertical line as point A. How many units to the left is point B translated?

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