Lesson 6-1: Explore Translations

Property

A translation is a rigid transformation that "slides" a figure across a plane to a new location. Every single point of the original figure (the pre-image) moves the exact same distance and in the exact same direction to create the new figure (the image). Because it is a rigid motion, the figure does not rotate, reflect, or change its size. Therefore, the pre-image and image are perfectly congruent and face the exact same way (they preserve orientation).

Examples

• Macro View: Sliding a physical ruler across your desk without rotating it.
• Micro Detail (Naming): When triangle ABC slides to a new position, the new triangle is named A'B'C' (read as "A prime, B prime, C prime"). Point A matches with A', B with B', and C with C'.
• Micro Detail (Direction): If you draw a straight line from A to A' and another from B to B', those lines will be perfectly parallel and the exact same length.

Explanation

While the property tells us the shape just "slides," here are the micro-details to watch out for:

1. Pre-image vs. Image: The original starting shape is called the "pre-image" (usually standard letters like A, B, C). The final landing spot is the "image" (indicated by the prime marks like A', B', C').
2. Congruence: Because it's a "rigid" motion, the pre-image and image are exactly identical. If the side length of AB was 5 units, the side length of A'B' is strictly 5 units. No stretching allowed!

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'!

Property

The visual slide of a translation is written as an algebraic coordinate rule: (x, y) -> (x + a, y + b).

• (x, y) represents any starting point on the pre-image.
• "a" is the horizontal shift added to the x-coordinate (use +a for Right, -a for Left).
• "b" is the vertical shift added to the y-coordinate (use +b for Up, -b for Down).

This single rule acts as a master instruction that applies identically to every point on the figure.

Examples

• Translating Words to Rule: "Left 4, Up 5" becomes (x, y) -> (x - 4, y + 5).
• Handling Zeroes: "Right 3, but no vertical movement" becomes (x, y) -> (x + 3, y). Notice we don't write y + 0, just y.
• Finding the Rule from Math: If M(2, 5) moves to M'(7, 1):
  • x-change: 7 - 2 = +5
  • y-change: 1 - 5 = -4
  • Rule: (x, y) -> (x + 5, y - 4).

Explanation

Let's look at the hidden mechanics of this formula:

1. The Arrow (->): This symbol means "maps to" or "becomes". It separates the "before" (x, y) from the "after" (x + a, y + b).
2. It's a Master Template: The rule (x, y) -> (x + 2, y - 3) doesn't mean x equals 2. It means "take whatever your starting x-coordinate is, and add 2 to it." You apply this identical template to every single vertex of your shape.