Geometric transformations are like magic tricks for shapes. They let us move, flip, and spin figures around the coordinate plane. We'll learn about translations, reflections, and rotations - each with its own special rules.
These transformations are powerful tools for understanding how shapes behave. We'll see how to apply them to figures, calculate new coordinates, and explore how they affect a shape's size, position, and orientation.
- Translations move every point of a figure the same distance in the same direction represented by a vector $(a, b)$, where $a$ is the horizontal shift and $b$ is the vertical shift
- Reflections flip a figure over a line called the line of reflection common lines include the $x$-axis, $y$-axis, and the lines $y = x$ and $y = -x$
- Rotations turn a figure around a fixed point called the center of rotation specified by an angle (in degrees) and a direction (clockwise or counterclockwise) common angles are 90°, 180°, and 270°
- To translate a figure by a vector $(a, b)$:
- Add $a$ to the $x$-coordinate of each point
- Add $b$ to the $y$-coordinate of each point
- To reflect a figure over a line:
- If reflecting over the $x$-axis, negate the $y$-coordinate of each point ($y$-coordinate becomes its opposite)
- If reflecting over the $y$-axis, negate the $x$-coordinate of each point ($x$-coordinate becomes its opposite)
- If reflecting over $y = x$, swap the $x$ and $y$ coordinates of each point ($(x, y)$ becomes $(y, x)$)
- If reflecting over $y = -x$, swap the $x$ and $y$ coordinates and then negate both coordinates ($(x, y)$ becomes $(-y, -x)$)
- To rotate a figure by an angle $\theta$ counterclockwise around the origin:
- For each point $(x, y)$, the new coordinates are $(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$
- For clockwise rotations, use the opposite angle ($-\theta$ instead of $\theta$)
- Apply the appropriate transformation rules to each point of the figure to determine the new coordinates
- For translations, add the vector components to the respective coordinates ($x + a$, $y + b$)
- For reflections, negate or swap coordinates based on the line of reflection ($-x$, $-y$, $(y, x)$, or $(-y, -x)$)
- For rotations, use the rotation formulas with the given angle and direction ($(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$ for counterclockwise, $(x \cos (-\theta) - y \sin (-\theta), x \sin (-\theta) + y \cos (-\theta))$ for clockwise)
- Translations preserve the size, shape, and orientation of the figure but change its position (shift the figure without altering its appearance)
- Reflections preserve the size and shape of the figure but change its orientation (create a mirror image across the line of reflection) and may change its position depending on the line of reflection
- Rotations preserve the size and shape of the figure but change its orientation based on the angle of rotation (turn the figure around the center of rotation) and may change its position depending on the center of rotation