Lesson 2.1

Vertical and Horizontal Translations

Moving graphs up, down, left, or right without changing their shape. Master the counter-intuitive "Inside is Opposite" rule.

Introduction

Prerequisite Connection: You observed in Lesson 1.5 that moved the graph DOWN. Today, we formalize exactly why that happened.

Today's Increment: We are learning the simplest "Rigid Transformations": shifts. The shape does not stretch or shrink; it merely relocates.

Why This Matters for Calculus: In Calculus, you won't always memorize the derivative of . Instead, you'll know the derivative of and use transformation rules to handle the shift.

Explanation of Key Concepts

Vertical Shifts (Outside)

Shift UP

If , add to the y-coordinate. Move UP.

Shift DOWN

If , subtract from the y-coordinate. Move DOWN.

Intuition: Vertical changes happen "after" the function does its job. It's an afterthought added to the output.

means "calculate height, then add 3."

Horizontal Shifts (Inside)

Shift LEFT

Addition inside moves LEFT (Negative direction).

Shift RIGHT

Subtraction inside moves RIGHT (Positive direction).

Warning: This is the "Inside is Opposite" rule.

means x must be 5 bigger to get back to 0. So we shift to +5 (Right).

Worked Examples

Level: Basic

Example 1: Vertical Shift

Graph relative to parent .

Analysis

The "+4" is outside the square. This is a vertical change.

Mapping

All points

become

(0,0) → (0,4)

Level: Intermediate

Example 2: Horizontal Shift

Graph relative to parent .

The Trap

We see "-3" and want to go left (negative). BUT it is inside.

Correct Move

"Inside is Opposite." Minus means RIGHT.

(0,0) → (3,0)

Level: Advanced (Calculus Prep)

Example 3: Combined Shift

Describe the transformation for .

Inside Change

Left 2

Outside Change

Down 5

Result

Vertex moves from (0,0) to (-2, -5).

Common Pitfalls

Mixing up Left and Right:
Students see and think "Positive is Right." You must memorize: Input changes are Counter-Intuitive.
Forgetting Order of Operations:
In , the shift is 3 to the right. But in , you must factor it first to see the true shift! (We cover this in Lesson 2.4).

Real-World Application

Computer Graphics: Moving Sprites

In video game development, a character is drawn around a local origin (0,0). To move a character across the screen, we apply translations to every pixel or vertex.

new_x = player.x + velocity_x
new_y = player.y + velocity_y

If the player presses "Right," we add to the X-coordinate. If they jump "Up," we add to the Y-coordinate. This is exactly in action.