Mastering Reflections and Rotations in Coordinate Planes

Reflections

Reflections create mirror images of shapes without changing their size. When you reflect a point, its distance from the reflection line stays the same, but it appears on the opposite side.

When reflecting across the x-axis, the y-coordinate changes sign while the x-coordinate remains the same: (x,y) → $x,-y$. For example, point (4,2) becomes $4,-2$. When reflecting across the y-axis, the x-coordinate changes sign: (x,y) → $-x,y$.

Reflections can also happen across diagonal lines. Reflecting across y=x swaps the coordinates: (x,y) → (y,x), turning $-1,3$ into $3,-1$. Reflecting across y=-x gives us (x,y) → $-y,-x$.

💡 Think of reflections like flipping a shape in a mirror - the shape stays the same size and form, but appears reversed across the line of reflection.

Rotations

Rotations involve turning shapes around a fixed point (usually the origin) by a specific angle. The shape maintains its size and form while changing position.

For a 90° clockwise rotation, use the formula (x,y) → $y,-x$. For example, $-5,2$ becomes (2,5). A 90° counterclockwise rotation transforms coordinates to $-y,x$, so (4,3) becomes $-3,4$.

A 180° rotation in either direction flips the signs of both coordinates: (x,y) → $-x,-y$. Point (2,10) becomes $-2,-10$. This is like turning something completely upside down.

For 270° rotations, clockwise gives $y,-x$ while counterclockwise gives $x,-y$. Remember that a 270° clockwise rotation is the same as a 90° counterclockwise rotation!

🔄 When rotating points, imagine them circling around the origin. Each 90° movement follows predictable patterns in how coordinates change.

Translations

Translations move shapes from one location to another without changing their size or orientation. Think of sliding a shape across the coordinate plane.

To translate a point, you simply add values to each coordinate using the formula $x+a, y+b$. The values of "a" and "b" tell you how far to move right/left and up/down.

For example, moving a shape 5 units right and 1 unit up means using the formula $x+5, y+1$. If we have point (0,12) and translate it 3 units right and 2 units down, we calculate $0+3, 12-2$ to get the new coordinates (3,10).

🏃‍♀️ When labeling translated shapes, use prime notation (like G') to show it's the same shape in a new position. This helps distinguish between the original and the transformed shape.