# Lesson 2

Transformations as Functions

• Let’s compare transformations to functions.

### 2.1: Math Talk: Transforming a Point

Mentally find the coordinates of the image of $$A$$ under each transformation.

• Translate $$A$$ by the directed line segment from $$(0,0)$$ to $$(0,2)$$.
• Translate $$A$$ by the directed line segment from $$(0,0)$$ to $$(\text-4,0)$$.
• Reflect $$A$$ across the $$x$$-axis.
• Rotate $$A$$ 180 degrees clockwise using the origin as a center.

### 2.2: Inputs and Outputs

1. For each point $$(x,y)$$, find its image under the transformation $$(x+12,y-2)$$.
1. $$A=(\text-10, 5)$$
2. $$B=(\text-4, 9)$$
3. $$C=(\text-2, 6)$$
2. Next, sketch triangle $$ABC$$ and its image on the grid. What transformation is $$(x,y) \rightarrow (x+12,y-2)$$?
3. For each point $$(x,y)$$ in the table, find $$(2x,2y)$$.
$$(x,y)$$ $$(2x,2y)$$
$$(\text-1, \text-3)$$
$$(\text-1, 1)$$
$$(5, 1)$$
$$(5, \text-3)$$
4. Next, sketch the original figure (the $$(x,y)$$ column) and image (the ($$2x,2y)$$ column). What transformation is $$(x,y) \rightarrow (2x,2y)$$?

### 2.3: What Does it Do?

1. Here are some transformation rules. Apply each rule to quadrilateral $$ABCD$$ and graph the resulting image. Then describe the transformation.
1. Label this transformation $$Q$$$$(x,y) \rightarrow (2x,y)$$
2. Label this transformation $$R$$$$(x,y) \rightarrow (x,\text-y)$$
3. Label this transformation $$S$$$$(x,y) \rightarrow (y,\text-x)$$

1. Plot the quadrilateral with vertices $$(4,\text-2),(8,4),(8,\text-6),$$ and $$(\text-6,\text-6)$$. Label this quadrilateral $$A$$.
2. Plot the quadrilateral with vertices $$(\text-2,4),(4,8),(\text-6,8),$$ and $$(\text-6,\text-6)$$. Label this quadrilateral $$A'$$.
3. How are the coordinates of quadrilateral $$A$$ related to the coordinates of quadrilateral $$A'$$?
4. What single transformation takes quadrilateral $$A$$ to quadrilateral $$A'$$?

### Summary

Square $$ABCD$$ has been translated by the directed line segment from $$(\text-1,1)$$ to $$(4,0)$$. The result is square $$A’B’C’D’$$.

$$A=(\text-1,1)$$ $$A’=(4,0)$$
$$B=(1,1)$$ $$B’=(6,0)$$
$$C=(1,\text-1)$$ $$C’=(6,\text-2)$$
$$D=(\text-1,\text-1)$$ $$D’=(4,\text-2)$$
$$Q=(\text-0.5,1)$$ $$Q’=(4.5, 0)$$
This table looks like a table that shows corresponding inputs and outputs of a function. A transformation is a special type of function that takes points in the plane as inputs and gives other points as outputs. In this case, the function’s rule is to add 5 to the $$x$$-coordinate and subtract 1 from the $$y$$-coordinate.
We write the rule this way: $$(x,y) \rightarrow (x+5, y-1)$$.