Function Transformations: Stretches and Compressions

May 25, 2024

Function Transformations: Horizontal and Vertical Stretches and Compressions

Introduction

  • Focus: horizontal and vertical stretches and compressions
  • Comparing f(x) to a modified form to see how values of "A" and "B" affect the graph

Vertical Stretches and Compressions

  • Equation: y = A
    • If A > 1: Vertical stretch by a factor of A
    • If 0 < A < 1: Vertical compression by a factor of A
  • Example with f(x) = x²
    • Original y-values: 1², 2², 3², 4²
    • For 2
      • y-values: 2 × 1, 2 × 4, 2 × 9, 2 × 16
      • Result: Vertical stretch
    • For ½
      • y-values: ½ × 1, ½ × 4, ½ × 9, ½ × 16
      • Result: Vertical compression

Key Points

  • Identify the value of A to find y-coordinates of the transformed function
  • Multiply each y-coordinate by "A", keeping x-coordinates the same
  • Example graphs:
    • Original function in blue
    • Vertically stretched function (h(x) = 2 f(x))
    • Vertically compressed function (g(x) = ½ x f(x))
  • Animation to visualize vertical stretches and compressions

Horizontal Stretches and Compressions

  • Equation: y = f(Bx)
    • If B > 1: Horizontal compress
    • If 0 < B < 1: Horizontal stretch
  • Example with f(x) = x²
    • Original y-values: 1, 4, 9, 16
    • For f(2x)
      • x-values: 1/2, 1, 3/2, 2
      • Result: Horizontal compression
    • For f(½x)
      • x-values: 2, 4, 6, 8
      • Result: Horizontal stretch

Key Points

  • Identify the value of B
  • For B > 1, divide original x-coordinates by B to get new x-values, y-coordinates remain unchanged
  • Example graphs:
    • Original function in blue
    • Horizontally compressed function (f(2x))
    • Horizontally stretched function (f(½x))
  • Animation to visualize horizontal stretches and compressions

Recognizing Parent Functions and Transformations

  • Recognize parent functions for f(x)

  • Example: f(x) = 3 |x|, identify parent function g(x) = |x|

    • V-shape, key points: (0, 0), (2, 2), (-2, 2)
    • Transform: f(x) = 3
    • Value of A = 3, vertical stretch
    • Points:
      • (0, 0) → (0 × 3, 0 × 3), (2, 2) → (2 × 3, 2 × 3), (-2, 2) → (-2 × 3, -2 × 3)
      • Result: Vertical stretch by factor of 3
  • Another Example: f(x) = √(2x)

    • Parent function g(x) = √x
    • Key points: (0, 0), (1, 1), (4, 2), (9, 3), (16, 4)
    • Transform: f(x) = √(2x)
    • Factor B = 2, horizontal compression
    • Points:
      • (0, 0) → (0 ÷ 2, 0), (1 ÷ 2, 1), (4 ÷ 2, 2), (9 ÷ 2, 3), (16 ÷ 2, 4)
      • Result: Horizontally compressed by factor of 2

Conclusion

  • Recap of vertical and horizontal stretches/compressions
  • Practical examples and key points to remember
  • Importance of recognizing parent functions and transformations “Vertical stretch or shrink occurs when the function is multiplied by a number. Horizontal stretch or shrink occurs when the input is multiplied by a number.”-andrews.edu