Recognizing Functions Using the Vertical Line Test

Introduction

• The vertical line test is a method used to determine if a graph represents a function.
• A graph represents a function if any vertical line drawn across the graph touches it at no more than one point simultaneously.

Examples and Analysis

Example 1: Linear Function

• Equation: y = x
• Graph: Straight line
• Vertical Line Test: Passes, as it touches the graph at only one point.
• Conclusion: The graph is a function.

Example 2: Parabola on Side

• Equation: x = y²
• Vertical Line Test: Fails, as it touches the graph at two points.
• Conclusion: The graph is not a function.

Example 3: Parabola

• Equation: y = x²
• Vertical Line Test: Passes, as it touches at only one point.
• Conclusion: The graph is a function.

Example 4: Circle

• Equation: x² + y² = 9
• Vertical Line Test: Fails, as it touches at two points.
• Conclusion: The graph is not a function.

Example 5: Horizontal Line

• Equation: y = 3
• Vertical Line Test: Passes, as it touches at only one point.
• Conclusion: The graph is a function.

Example 6: Vertical Line

• Equation: x = 4
• Vertical Line Test: Fails, as it touches at infinite points.
• Conclusion: The graph is not a function.

Additional Graphs

Square Root and Absolute Value Functions

• Graphs:
  • Square root of x: Passes the vertical line test.
  • Absolute value of x (V shape): Passes the vertical line test.
• Conclusion: Both are functions.

Sine and a Non-Function Graph

• Graph 1: (Unnamed non-function)
  • Fails the vertical line test, touches at three points.
  • Conclusion: Not a function.
• Graph 2: Sine function
  • Passes the vertical line test.
  • Conclusion: Is a function.

Cubic and Cube Root Functions

• Graphs:
  • x³: Passes the vertical line test.
  • Cube root of x: Passes the vertical line test.
• Conclusion: Both are functions.

Final Example

• Graph: (Unnamed non-function)
  • Fails the vertical line test, touches at two points.
  • Conclusion: Not a function.

Summary

• The vertical line test is a useful tool to check if a graph represents a function.
• Understanding how to apply this test helps in identifying whether equations represent functional relationships.