Lesson 4.3

The Vertical Line Test

The fastest way to spot a fake. One vertical line is all it takes to expose a relation that isn't a function.

Introduction

In Lesson 4.2, we learned that a function cannot have repeating inputs (x-values). On a graph, an x-value is a location left-to-right. If a graph has two points at the same x-location, they will be stacked vertically.

Past Knowledge

Lesson 4.2 (Definition of Function). We are just visualizing that definition now.

Today's Goal

Use a vertical line to instantly determine if a graph is a function.

Future Success

Circles and Ellipses fail this test, which is why their equations look different ($x^2 + y^2 = r^2$).

Key Concepts

The Pencil Scan

Imagine sliding a vertical line (like a pencil) across the graph from left to right.

PASSED (Function)

The line never touches more than one point at a time.

FAILED (Not a Function)

The line hits two points simultaneously.

Worked Examples

Example 1: Linear Graph

Basic

Is this line a function?

No matter where you draw a vertical line, it only crosses the blue line once. YES, it is a function.

Example 2: The Circle

Intermediate

Is a circle a function?

If we draw a line at , it hits the top at (0,3) and the bottom at (0,-3). Two hits. NO, not a function.

Example 3: Discrete Points

Advanced

Is this scatter plot a function?

Look closely at x=2. There are two points stacked on top of each other: (2,1) and (2,4). The test fails. NO, not a function.

Common Pitfalls

The Horizontal Line

Students sometimes scan horizontally (left-to-right). A horizontal line hitting twice gives you clues about "One-to-One" functions, but it does NOT mean it's not a function. Only Vertical matters for definition.

Real-Life Applications

Stock Prices: A stock price chart is a function. At any specific moment in time (x), a stock has exactly one price (y). It can't be $50 and $100 at the exact same second. If the line went vertical, the market would break.