Skill: Describe transformations using coordinates

A transformation changes every point in a figure using the same coordinate rule. Apply the rule to each vertex, then connect the new points in the same order.

Coordinate rules

• Translation right $a$ units and up $b$ units:

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

• Reflection across the $x$-axis:

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

• Reflection across the $y$-axis:

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

• Rotation $90^\circ$ counterclockwise about the origin:

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

• Rotation $180^\circ$ about the origin:

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

• Rotation $270^\circ$ counterclockwise about the origin:

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

For a rotation, always switch the positions of $x$ and $y$ as shown. Then check whether either coordinate changes sign.

Worked example

Triangle $ABC$ has vertices

$$A(1,2),\qquad B(4,2),\qquad C(1,5).$$

Rotate the triangle $90^\circ$ counterclockwise about the origin. Find the coordinates of the image.

Use the rule

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

Apply it to each vertex:

$$\begin{aligned} A(1,2)&\rightarrow A'(-2,1),\\ B(4,2)&\rightarrow B'(-2,4),\\ C(1,5)&\rightarrow C'(-5,1). \end{aligned}$$

Therefore, the rotated triangle has vertices

$$\boxed{A'(-2,1),\ B'(-2,4),\ C'(-5,1)}.$$

The transformation is described by saying: “Each point is rotated $90^\circ$ counterclockwise about the origin, using the coordinate rule $(x,y)\rightarrow(-y,x)$.”