Transformations in the Cartesian Plane Study Guide

Describe the position of points in the Cartesian plane.
Describe transformations, including translations, reflections, and rotations, using coordinates in the Cartesian plane.
Transformation is defined as a change in any form or appearance. It is an operation changing one configuration into another according to a mathematical rule (e.g., mapping, rotation, or change of variables/coordinates).
Pre-image: The initial image before a transformation occurs.
Image: The final image resulting from a transformation mapping the pre-image.

Origin and Naming: The Cartesian plane is also referred to as the rectangular coordinate plane. It is named after Rene Descartes, a French mathematician, scientist, and philosopher who introduced the coordinate system in the 1600s.
Structure: It is formed by combining a horizontal number line ( $x$ -axis) and a vertical number line ( $y$ -axis).
Components:
- $x$ -axis: The horizontal number line.
- $y$ -axis: The vertical number line.
- Origin: The point where the $x$ -axis and $y$ -axis intersect, denoted by the coordinates $(0, 0)$ .
- Ordered Pair $(x, y)$ : Represents a specific location (point) on the plane. For example, the point $(-3, 1)$ indicates a location $3$ units to the left and $1$ unit up from the origin.
Quadrants: The axes divide the plane into four regions called quadrants, numbered counterclockwise:
- Quadrant I: Top right (Positive $x$ , Positive $y$ ).
- Quadrant II: Top left (Negative $x$ , Positive $y$ ).
- Quadrant III: Bottom left (Negative $x$ , Negative $y$ ).
- Quadrant IV: Bottom right (Positive $x$ , Negative $y$ ).
Example Plots:
- $A(1, 5)$ : Quadrant I
- $B(7, 0)$ : $x$ -axis
- $C(-1, 3)$ : Quadrant II
- $D(-5, -9)$ : Quadrant III
- $E(2, -4)$ : Quadrant IV
- $F(0, 1)$ : $y$ -axis
- $G(-8, 6)$ : Quadrant II
- $H(6, 10)$ : Quadrant I
- $I(-7, -7)$ : Quadrant III
- $J(0, 0)$ : Origin

Definition: A change that moves all points of a figure the same distance in the same direction. It is essentially a slide in a straight line without turning.
Real-world Examples: Footprints in the sand, rows of leaves, decorative border patterns.
Coordinate Rules:
- Translation moves can be expressed as $(x, y) \rightarrow (x+h, y+k)$ .
- Example: A translation of 4 units left and 1 unit down is represented by $(x, y) \rightarrow (x - 4, y - 1)$ .
- Given points: $X(3, 3), Y(4, 0), Z(2, -4), W(0, 1)$ . Under the rule $4$ units left and $1$ unit down, the new points are $X'(-1, 2), Y'(0, -1), Z'(-2, -5), W'(-4, 0)$ .
- Example Rule: $(x, y) \rightarrow (x + 3, y - 5)$ for points $C(-5, 2), A(-3, 4), T(0, 0)$ results in $C'(-2, -3), A'(0, -1), T'(3, -5)$ .

Definition: A flip of a figure over a "mirror" (line of symmetry). The line of reflection divides the shape into two matching halves.
Real-world Examples: Butterfly wings, mountain reflections in a lake, symmetrical face or designs.
Procedure to Reflect a Point:
1. Identify the line of reflection.
2. Measure the distance of the point from the line.
3. Mark the point on the opposite side at the same distance.
4. Connect the new vertices to form the image.
Standard Rules:
- Across $x$ -axis ( $y = 0$ ): $(x, y) \rightarrow (x, -y)$ . The $x$ -coordinate stays the same; the $y$ -coordinate changes sign.
- Across $y$ -axis ( $x = 0$ ): $(x, y) \rightarrow (-x, y)$ . The $y$ -coordinate stays the same; the $x$ -coordinate changes sign.
- Across line $y = x$ : $(x, y) \rightarrow (y, x)$ . The coordinates switch places.
- Across line $y = -x$ : $(x, y) \rightarrow (-y, -x)$ .
- Across vertical line $x = a$ : $(x, y) \rightarrow (2a - x, y)$ . Example: Reflecting $K(1, 3)$ across $x = -3$ results in $K'(-7, 3)$ .
- Across horizontal line $y = b$ : $(x, y) \rightarrow (x, 2b - y)$ .

Definition: A change that turns every point of a pre-image through a specified angle and direction about a fixed point called the center of rotation.
Real-world Examples: Pinwheels, sunflower seed spirals, wheels, windmills, clocks, dancers, and starfish.
Rules for Rotation about the Origin $(0, 0)$ :
- 180^\round rotation: $(x, y) \rightarrow (-x, -y)$ . Coordinates keep their position but take opposite signs.
- 90^\round clockwise (or 270^\round counterclockwise): $(x, y) \rightarrow (y, -x)$ . Interchange coordinates and adjust signs based on the new quadrant.
- 90^\round counterclockwise: $(x, y) \rightarrow (-y, x)$ .

Definition: A change in the size of a figure that requires a center point and a scale factor. The shape remains the same (similar), but the size changes.
Real-world Examples: Seashells, succulents, or lotus flowers at different scales; enlarging geometric patterns.
Scale Factor ( $k$ ): The ratio of the size of the new image to the old image.
- Enlargement: If scale factor k > 1, the image stretches/grows.
- Reduction: If scale factor 0 < k < 1, the image shrinks.
- Congruency: If scale factor $k = 1$ , the original image and produced image are congruent.
Coordinate Rule (Centered at Origin):
- $(x, y) \rightarrow (kx, ky)$ .
- Example: Dilation with Scale Factor $2$ for points $A(0, 2), B(2, -1), C(-2, -2)$ results in $A'(0, 4), B'(4, -2), C'(-4, -4)$ .
- Example: Dilation with Scale Factor $\frac{1}{2}$ for points $B(-2, 4), C(4, 4), D(2, -2), A(-4, 2)$ results in $B'(-1, 2), C'(2, 2), D'(1, -1), A'(-2, -1)$ .

A. Matching Type:
1. Rotation: A figure turns around a fixed point.
2. Translation: A figure slides without turning.
3. Dilation: A figure changes size but keeps the same shape.
4. Reflection: A figure flips across a line.
B. Coordinate Tasks:
1. $A(2, 3)$ translated $5$ units left and $1$ unit down: $A'(-3, 2)$ .
2. $B(-4, 2)$ reflected across the $x$ -axis: $B'(-4, -2)$ .
3. $C(1, -3)$ rotated 90^\round counterclockwise about the origin: $C'(3, 1)$ .
4. $D(2, 1)$ dilated by scale factor $3$ centered at origin: $D'(6, 3)$ .