Lesson 2.2

Reflections Across the Axes

Flipping graphs over the x-axis or y-axis. Learn where the negative sign goes and what it does.

Introduction

Prerequisite Connection: You know that is the opposite of . Geometrically, this means flipping a point to the other side of zero. Today, we apply this logic to entire shapes.

Today's Increment: We are learning the second type of "Rigid Transformation": reflections. A reflection creates a mirror image of the graph.

Why This Matters for Calculus: Many integration problems (area under a curve) rely on symmetry. Recognizing that a graph is a reflection helps simplify complex area calculations to zero or double specific values.

Explanation of Key Concepts

Reflection over X-Axis (Vertical Flip)

The negative is OUTSIDE. It negates the output (y-value). Positive heights become negative heights.

(x, y) → (x, -y)

X-AXIS

Reflection over Y-Axis (Horizontal Flip)

The negative is INSIDE. It negates the input (x-value). Left becomes right, right becomes left.

(x, y) → (-x, y)

Y-AXIS

Worked Examples

Level: Basic

Example 1: Vertical Reflection

Graph relative to .

Analysis

The negative is outside. The "cup" flips upside down.

Mapping

All positive outputs become negative.

(2, 4) → (2, -4)

Level: Intermediate

Example 2: Horizontal Reflection

Graph relative to .

Visualizing

Usually, square root shoots to the RIGHT. Now, it shoots to the LEFT.

Domain Check

Normally

. Now

, which means

. The domain flipped!

Level: Advanced (Calculus Prep)

Example 3: Double Reflection

Graph .

Inside Negative

Flip Horizontally (Left)

Outside Negative

Flip Vertically (Down)

Result

The graph is in Quadrant III (Bottom-Left).

Common Pitfalls

Reflecting EVEN Functions:
If you reflect across the y-axis, it looks identical. Students think "I did it wrong." No, the graph is just symmetric!
Order of Operations with Shifts:
Reflecting and shifting are NOT commutative. is "Left 2, then Flip." is "Flip, then Up 2."

Real-World Application

Optics: Mirrors and Lenses

The entire field of optics is based on reflections. When you look in a mirror, your image is a "Y-Axis Reflection" of yourself (Left becomes Right).

In cameras, the lens often flips the image upside down (X-Axis Reflection) onto the sensor. The camera software must digitally flip it back () so you see the picture correctly.