Lesson 5.14

Reflections

Looking in the mirror flips everything. In algebra, a negative sign does the exact same thing.

Introduction

We learned how to slide graphs up and down. Now we learn how to flip them over. A Reflection creates a mirror image of the graph across an axis.

Past Knowledge

Lesson 5.1 (Slope). Remember that positive slope goes up, and negative slope goes down. They are related!

Today's Goal

Understand how reflects a graph over the x-axis.

Future Success

This negative sign rule () is why some parabolas open downwards like a frown.

Key Concepts

Reflection over the X-Axis

The Rule

Multiply the ENTIRE function by -1.

Example: .

Visual Effect

Positive becomes Negative .

Top flips to Bottom.

Worked Examples

Example 1: Basic Reflection

Basic

Reflect over the x-axis.

Step 1: Apply Negative

Put a negative sign in front of everything.

Analysis

The slope changed from positive 3 to negative 3. It now goes downhill.

Example 2: Distributing the Negative

Intermediate

Reflect .

Step 1: Parentheses

Write .

Step 2: Distribute

Flip signs of BOTH terms.

Notice: The Y-intercept also flipped from +2 to -2.

Example 3: Double Negative

Advanced

Reflect .

Step 1: Negate

-(-2x - 5)

Step 2: Simplify

Negative times Negative is Positive.

Common Pitfalls

Forgetting to Distribute

When reflecting , many students write . This is wrong! You must flip the 5 too. The correct answer is .

Confusing with Y-Axis Reflection

Reflecting over the Y-axis () is a different rule. It flips left/right. Today we are only doing (Up/Down).

Real-Life Applications

Business Profit vs Loss: If models your profit, then models your loss. If you made \$100 (), reflecting means you lost \$100 (). The shape of the curve is the same, just flipped.