Overview

This topic explores how functions are represented graphically and how graphs are affected by various transformations. Mastering graph behavior and modifications helps build visual intuition for solving equations and modeling real-world problems.

Key Concepts and Transformations

• Parent Functions: Basic graphs like f(x) = x, x^2, |x|, \sqrt{x}, 1/x, 2^x, \log(x), and \sin(x)
• Vertical Shifts: f(x) + c shifts up, f(x) - c shifts down
• Horizontal Shifts: f(x - c) shifts right, f(x + c) shifts left
• Reflections:
  • -f(x) reflects over the x-axis
  • f(-x) reflects over the y-axis
• Stretches/Compressions:
  • af(x) stretches vertically if |a| > 1; compresses if 0 < |a| < 1
  • f(bx) compresses horizontally if |b| > 1; stretches if 0 < |b| < 1
• Even/Odd Functions:
  • Even: symmetric about y-axis, f(-x) = f(x)
  • Odd: symmetric about origin, f(-x) = -f(x)

Step-by-Step Example

Problem: Describe the transformations of f(x) = -2(x - 3)^2 + 1 based on the parent function f(x) = x^2

Step 1: (x - 3) → horizontal shift right 3 units

Step 2: -2 → reflect over x-axis and vertical stretch by factor of 2

Step 3: +1 → vertical shift up 1 unit

Answer: Right 3 units, reflect over x-axis, stretch vertically by 2, up 1 unit

Quick Tip