Lecture Notes: Graphing Transformations of a Function

Introduction

• Discussing graphing transformations of a given function using the graph of the function.
• Four types of transformations will be covered.

Example 1: Vertical Shift

• Function: f(x) (in red on graph).
• Transformation: g(x) = f(x) + 3.
• Key Points: Identify three key points on f(x).
• Procedure: Add 3 to each y-coordinate.
  • Shift graph up by 3 units.
• Examples:
  • (−3, 1) ➔ (−3, 4)
  • (−1, 2) ➔ (−1, 5)
  • (3, −2) ➔ (3, 1)

Example 2: Reflection over x-axis

• Function: f(x) (same red graph).
• Transformation: g(x) = -f(x).
• Procedure: Change the sign of each y-coordinate.
  • Reflect graph across x-axis.
• Examples:
  • (−3, 1) ➔ (−3, −1)
  • (−1, 2) ➔ (−1, −2)
  • (3, −2) ➔ (3, 2)

Example 3: Horizontal Shift

• Function: f(x) (same red graph).
• Transformation: g(x) = f(x + 3).
• Understanding: Not adding 3 to x-coordinates directly.
  • Subtract 3 from x-coordinates.
  • Graph shifts left by 3 units.
• Examples:
  • (−3, 1) ➔ (−6, 1)
  • (−1, 2) ➔ (−4, 2)
  • (3, −2) ➔ (0, −2)
• Notes:
  • For g(x) = f(x + c):
    • If c < 0 (f(x - c)), shift right c units.
    • If c > 0, shift left c units.

Example 4: Reflection over y-axis

• Function: f(x) (same red graph).
• Transformation: g(x) = f(−x).
• Procedure: Change the sign of each x-coordinate.
  • Reflect graph across y-axis.
• Examples:
  • (−3, 1) ➔ (3, 1)
  • (−1, 2) ➔ (1, 2)
  • (3, −2) ➔ (−3, −2)

Conclusion

• Reviewed four main types of transformations: vertical shift, reflection over x-axis, horizontal shift, and reflection over y-axis.
• Important to correctly identify how each transformation affects the graph.