Unit Activity: Introduction to Geometry and Transformations Answers

Task 1 answers

A. Coordinate rules

B. How these functions differ from familiar algebra functions

A coordinate transformation takes an ordered pair as its input and returns another ordered pair:

$$T:\mathbb{R}^2\to\mathbb{R}^2.$$

In many earlier algebra examples, a function accepts one real number and returns one real number, such as $f:\mathbb{R}\to\mathbb{R}$. A geometric transformation still follows the function rule—each input point has exactly one image—but both coordinates are transformed together.

C. Composite transformations

Apply transformations in the stated order.

1. Rotate $90^\circ$ counterclockwise, then reflect across the $x$-axis:

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

1. Reflect across the $y$-axis, then translate $a$ units right and $b$ units up:

$$(x,y)\mapsto(-x,y)\mapsto(-x+a,y+b).$$

1. Translate $a$ units right and $b$ units up, rotate $180^\circ$, then reflect across the $y$-axis:

$$(x,y)\mapsto(x+a,y+b) \mapsto(-x-a,-y-b) \mapsto(x+a,-y-b).$$

The order matters. For example, reflecting first and translating second generally produces a different image from translating first and reflecting second.

Conclusion

Treat every transformation as a function on coordinate pairs. Write each rule, pass the output of one rule into the next, and simplify only after all operations have been applied.