Lesson 11: Graphing Transformations

New Concept

Transformations are operations on a geometric figure that alter its position or size.

What’s next

This is just our starting point. You'll soon work through examples showing how to reflect, rotate, translate, and resize figures on the coordinate plane.

Property

A reflection occurs across a line. A segment between corresponding points of a figure and its reflection is perpendicular to the line of reflection. If we were to fold a graph along the line of reflection, the figures would align exactly.

Examples

Reflecting $\triangle ABC$ with $A(-2, 6)$ and $B(-5, 2)$ across the x-axis results in $A'(-2, -6)$ and $B'(-5, -2)$.
When reflecting a point like $(4, 3)$ across the y-axis, the new coordinates become $(-4, 3)$.

Explanation

Imagine a mirror on a line! Your shape's reflection appears on the other side, exactly the same distance away. If you fold the paper on the mirror line, the two shapes would match up perfectly, creating a symmetrical image. It is a flip without changing the size or shape.

Property

A positive rotation turns a figure counterclockwise about (around) a point. The point of rotation is fixed, and the figure spins around it. The path of any point during the rotation sweeps out an arc of the specified angle.

Examples

Rotating a point $(5, 2)$ $90^\circ$ counterclockwise about the origin gives a new point at $(-2, 5)$.
Rotating a point $(5, 2)$ $180^\circ$ counterclockwise about the origin results in a new point at $(-5, -2)$.

Explanation

Think of it as spinning a shape around a fixed point, like a pin on a board! The shape pivots counterclockwise by a set angle, without changing its size or form. The whole figure moves together in a circular path, like riders on a Ferris wheel turning around the center axle.

Property

A translation on a coordinate grid is fully described by combining a horizontal shift and a vertical shift. The horizontal shift tells you how far left or right to move parallel to the x-axis (Right is a positive shift, Left is a negative shift). The vertical shift tells you how far up or down to move parallel to the y-axis (Up is a positive shift, Down is a negative shift). Together, these two directions describe the exact diagonal path of the entire figure.

Examples

• Step-by-Step Shift: To describe how P(1, 2) moves to P'(4, 0):
  1. Horizontal first: Start at x=1, count right to x=4. That is "Right 3".
  2. Vertical second: Start at y=2, count down to y=0. That is "Down 2".
  3. Description: "Translated 3 units right and 2 units down."

Explanation

When describing shifts, students often make these small but critical mistakes:

1. Counting Grid Lines, Not Points: When counting the distance from A to A', do not count the dot you start on as "1". You only count the "jumps" or "spaces" between the grid lines.
2. Order Matters: Always describe the Left/Right (horizontal) movement first, followed by the Up/Down (vertical) movement. This builds the habit needed for writing (x, y) coordinate rules later.
3. Connecting the right dots: Always count from A to A'. Do not accidentally count from A to B'!