Rotate Figures and Write Transformation Rules

Start with one point and one turn: rotate $(3, 1)$ by $90^\circ$ counterclockwise about the origin.

Think of the path from the origin to $(3, 1)$ as two straight steps: $3$ units right, along the horizontal $x$-axis, then $1$ unit up, parallel to the vertical $y$-axis. The rotation turns the whole path as one rigid piece, so each step keeps its length and turns $90^\circ$ counterclockwise.

• A step right becomes a step up. The half of the $y$-axis that points up is exactly $90^\circ$ counterclockwise from the half of the $x$-axis that points right.
• A step up becomes a step left. The half of the $x$-axis that points left is exactly $90^\circ$ counterclockwise from the half of the $y$-axis that points up.

So the turned path is $3$ units up, then $1$ unit left, and it ends at $(-1, 3)$.

The figure shows both paths with verified coordinates. The steps to Point G at $(3, 1)$ are two sides of the triangle with corners $(0, 0)$, $(3, 0)$ and $(3, 1)$. The turned steps to Point H at $(-1, 3)$ are two sides of the triangle with corners $(0, 0)$, $(0, 3)$ and $(-1, 3)$.

The same reasoning works for any point. Let $x$ and $y$ stand for its coordinates in grid units. Each can be any number: positive, negative or zero. The path from the origin to $(x, y)$ is $x$ units right, then $y$ units up. After a $90^\circ$ counterclockwise turn about the origin, the path is $x$ units up, then $y$ units left. So the image has $x$-coordinate $-y$ and $y$-coordinate $x$.

You can write this in one line, called a rule: $(x, y) \to (-y, x)$. Read the arrow as goes to. The symbol $-y$ means the opposite of $y$, the same number with its sign changed. This rule belongs to one turn only: $90^\circ$ counterclockwise about the origin.

The rule holds whatever the signs of $x$ and $y$. When $x$ is negative, the first step goes left, and the turn carries a step left to a step down, so the image's $y$-coordinate is still $x$. When $y$ is negative, the second step goes down, and the turn carries a step down to a step right, so the image's $x$-coordinate is still $-y$.

So far you have used rules. Now you will write the rule for a transformation: one line with an arrow that tells where every point $(x, y)$ goes.

If you know what the transformation does, put it into coordinates.

• Reflecting across the $x$-axis keeps $x$ and changes $y$ to its opposite: $(x, y) \to (x, -y)$.
• Reflecting across the $y$-axis changes $x$ to its opposite and keeps $y$: $(x, y) \to (-x, y)$.
• A translation slides every point the same distance in the same direction. Sliding $3$ units right and $2$ units down adds $3$ to $x$ and subtracts $2$ from $y$: $(x, y) \to (x + 3, y - 2)$.
• A $180^\circ$ rotation about the origin changes both $x$ and $y$ to their opposites: $(x, y) \to (-x, -y)$.

Sometimes you get coordinates instead of a description: each vertex of a figure matched with the vertex of the image it lands on. These matched pairs are called corresponding vertices. Then you compare the pairs, write the pattern you find using $x$ and $y$, and test it on every pair with exact coordinates.

Takeaway. A rotation turns a figure about a fixed center without changing its size or shape. When the center is the origin and the turn is $90^\circ$, $180^\circ$ or $270^\circ$, the turn changes the coordinates of every point in one fixed way, so you can rotate a figure exactly by applying the rule for that turn to each vertex. A rule such as $(x, y) \to (-y, x)$ speaks for every point at once. When you write a rule yourself, make sure it fits every pair of corresponding vertices, using exact coordinates and not the look of a drawing.

Next, you will describe a transformation from a figure and its image, verify that these transformations keep lengths and angle measures the same and keep parallel lines parallel, and find and correct errors in graphed transformations.