Overview

This topic covers how to graph basic functions and apply transformations to those graphs. Students will learn how translations, reflections, stretches, and compressions modify the appearance of function graphs. Understanding graph behavior helps with interpreting equations and modeling real-world situations.

Key Concepts and Structures

• Basic Parent Functions: Include f(x) = x, x^2, |x|, \sqrt{x}, and 1/x.
• Translations: Shift graphs up/down and left/right.
  - f(x) + c: Up
  - f(x) - c: Down
  - f(x + c): Left
  - f(x - c): Right
• Reflections: Flip graphs across axes.
  - -f(x): Reflect over x-axis
  - f(-x): Reflect over y-axis
• Stretching and Compressing:
  - af(x): Vertical stretch if a > 1; compression if 0 < a < 1
  - f(bx): Horizontal compression if b > 1; stretch if 0 < b < 1
• Intercepts: Points where the graph crosses axes. Set y = 0 for x-intercept, x = 0 for y-intercept.
• Symmetry: - Even functions: symmetric about y-axis f(-x) = f(x)
  - Odd functions: symmetric about origin f(-x) = -f(x)
• Asymptotes: Lines the graph approaches but never touches (often in rational or exponential functions).

Step-by-Step Example

Problem: Describe the transformation of f(x) = -(x - 2)^2 + 3 from the parent function f(x) = x^2.

Step 1: Recognize structure: f(x) = a(x - h)^2 + k is vertex form.

Step 2: Identify the transformations:
- (x - 2): Shift right 2
- + 3: Shift up 3
- - sign: Reflect over x-axis

Final Answer: Reflect over the x-axis, shift right 2 units, and up 3 units.

Quick Tip