Mastering Reflections and Rotations in Coordinate Planes

Jazmin M

12/7/2025

Math (SAT®)

Understanding Reflection and Rotation in Coordinate Planes

Subjects

Math (SAT®)

Heart of Algebra

Graphing Linear Equations

•

Dec 7, 2025

•

3 pages

Mastering Reflections and Rotations in Coordinate Planes

Jazmin M

@jazminm_eufa

Transformations in geometry are like giving shapes different instructions on... Show more

1 / 3

REFLECTION
Definition
(a mirror image)
not a change in zize
and shope
AAAAAAAAAAAAA
X values stay the same
y values become opposite
↑
Across

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.

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Subjects

Math (SAT®)

Heart of Algebra

Graphing Linear Equations

Math (SAT®)

•

Dec 7, 2025

•

3 pages

Mastering Reflections and Rotations in Coordinate Planes

Jazmin M

@jazminm_eufa

Transformations in geometry are like giving shapes different instructions on how to move around a coordinate plane. Understanding reflections, rotations, and translations helps you see how shapes can be manipulated while maintaining their core properties.

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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.

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.

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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.

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.

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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.

🏃‍♀️ 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.