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

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