Lesson 4.1.3

Translations on the Coordinate Plane

Take vector translations from Lesson 4.1.2 and apply them on the coordinate plane — graphing both the pre-image and image and writing formal algebraic mapping rules.

Introduction

In Lesson 4.1.2 you learned vector notation. Now you'll combine that knowledge with the coordinate plane — plotting the original figure, applying the rule, and graphing the translated image. This is the “hands-on” lesson that solidifies translations.

Past Knowledge

Vectors & component form (4.1.2). Plotting on the coordinate plane (1.2).

Today's Goal

Graph translations on the coordinate plane and write/apply algebraic mapping rules.

Future Success

All coordinate-plane transformations (reflections, rotations) follow this same workflow.

Key Concepts

Algebraic Mapping Rule

Add the horizontal component to every -coordinate and the vertical component to every -coordinate.

Graphing Workflow

  1. Plot the pre-image on the coordinate plane.
  2. Apply the rule to each vertex: add to , add to .
  3. Plot the image vertices and connect them in order.
  4. Verify: each image segment should be parallel to and the same length as its corresponding pre-image segment.

Determining the Rule from a Graph

If you're given the graph of a pre-image and its translated image:

  1. Pick a vertex and its image (e.g., and ).
  2. Compute and .
  3. Write the rule: .

Worked Examples

Basic

Applying a Rule

Translate with , , by the rule .

Vector
⟨,⟩
Adjust the vector to explore
R(1,3)→R′(-2,7)
S(4,5)→S′(1,9)
T(2,-1)→T′(-1,3)
Intermediate

Finding the Rule from Pre-image & Image

and . What is the mapping rule?

Using :

Verify with : ,

Vector
⟨,⟩
Adjust the vector to explore
A(3,-2)→A′(-1,4)
B(5,0)→B′(1,6)
Advanced

Finding a Pre-image Point

Under the translation , the image of a point is . Find the pre-image .

Work backwards — subtract the vector components:

Vector
⟨,⟩
Adjust the vector to explore
P(3,4)→P′(8,1)

Common Pitfalls

Applying the Rule to Only One Coordinate

Both and must be updated. Students sometimes shift only or only and wonder why the image looks wrong.

Wrong Direction When Finding the Pre-image

If the rule is , finding the pre-image means subtracting 5 from and adding 3 to . Many students accidentally apply the rule forward again.

Real-Life Applications

Scrolling on Your Phone

When you scroll a webpage, the entire content translates vertically on the screen — every pixel shifts by the same vector. The same math runs behind pinch-to-zoom with a pan.

Assembly Lines

In manufacturing, conveyor belts apply a continuous translation to every object — moving each item the same distance in the same direction along the production line.

Practice Quiz

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