Overview

This topic explores how functions can be represented graphically, symbolically, numerically, and descriptively. You'll learn how to interpret and analyze graphs, identify transformations, and recognize symmetry and intercepts. Understanding these representations supports better interpretation of mathematical models.

Key Concepts and Structures

• Graph of a Function: A visual representation showing all (x, y) points that satisfy the function.
• Intercepts:
  • x-intercept: Point(s) where the graph crosses the x-axis (y = 0)
  • y-intercept: Point where the graph crosses the y-axis (x = 0)
• Symmetry:
  • Y-axis symmetry: even function
  • Origin symmetry: odd function
• Translations: Move a graph horizontally or vertically.
  - f(x) + c: shifts up
  - f(x - c): shifts right
• Reflections: Flip a graph across an axis.
  - -f(x): reflects over x-axis
  - f(-x): reflects over y-axis
• Domain and Range:
  • Domain: Set of possible x-values
  • Range: Set of resulting y-values
• Multiple Representations: Functions may be described through tables, equations, graphs, or word problems. Recognizing the connection between these forms is key.

Step-by-Step Example

Problem: Given f(x) = (x - 1)^2, describe the graph and list its key features.

Step 1: Identify the parent function: f(x) = x^2 is a parabola opening upward.

Step 2: Analyze transformation: (x - 1)^2 means the graph is shifted right by 1 unit.

Step 3: State features:
- Vertex at (1, 0)
- Opens upward
- Axis of symmetry: x = 1
- Minimum value: y = 0
- Domain: all real numbers
- Range: y ≥ 0

Final Summary: This is a parabola shifted right 1 unit with its vertex at (1, 0) and a minimum value of 0.

Quick Tip