Describe transformations using coordinate rules

A coordinate rule describes how every point $(x,y)$ and the relationships among points change under a transformation: translations add signed horizontal and vertical amounts, reflections change appropriate coordinate signs, and rotations about the origin interchange coordinates and/or change their signs. Learners interpret these rules to predict image coordinates and connect algebraic changes with preserved distances, angle measures, and congruence. The scope excludes generalized transformations about arbitrary centers and formal matrix representations.

Detailed Explanation: Describe transformations using coordinate rules

A coordinate rule tells how to change every point in a figure to find its image.

Translation right $a$ and up $b$ :

(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)

The new coordinates must be found by applying the rule to both coordinates of every point.

Example: Triangle $ABC$ has vertices

A(1,4),\qquad B(3,4),\qquad C(1,2).

It is translated $3$ units right and $2$ units down.

Step 1: Write the coordinate rule

Moving $3$ units right adds $3$ to the $x$ -coordinate. Moving $2$ units down subtracts $2$ from the $y$ -coordinate.

So the rule is

(x,y)\rightarrow (x+3,y-2).

Step 2: Apply the rule to each vertex

For $A(1,4)$ :

A'(1+3,4-2)=(4,2).

For $B(3,4)$ :

B'(3+3,4-2)=(6,2).

For $C(1,2)$ :

C'(1+3,2-2)=(4,0).

Therefore, the image triangle has vertices

\boxed{A'(4,2),\quad B'(6,2),\quad C'(4,0)}.

Step 3: Check the meaning

Every point moved the same distance and in the same direction: $3$ units right and $2$ units down. Therefore, side lengths and angle measures stay the same, so the original triangle and its image are congruent.

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Cartesian Grid - Rotation of Point (Grid to Coordinates) around Origin

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