Lecture on Functions and Transformations

Introduction

• Discussion on functions and their transformations.
• Focus on vertical and horizontal shifts, stretches, and reflections.

Types of Shifts

Vertical Shifts

• f(x) + 2: Vertical shift up by 2 units.
• f(x) - 3: Vertical shift down by 3 units.

Horizontal Shifts

• f(x - 4): Horizontal shift right by 4 units.
• f(x + 3): Horizontal shift left by 3 units.

Reflections

• Negative outside: Reflects over the x-axis.
• Negative inside: Reflects over the y-axis.
• Negative inside and outside: Reflects over the origin.

Stretches and Shrinks

Vertical Changes

• 2f(x): Vertical stretch.
• (1/2)f(x): Vertical shrink.

Horizontal Changes

• f(2x): Horizontal shrink.
• f((1/2)x): Horizontal stretch.

Examples of Transformations

Quadratic Functions

• f(x) = x² (Parent Function): Parabola.
• x² + 3: Shift up 3 units - Starting point at (0, 3).
• x² - 2: Shift down 2 units.

Absolute Value Functions

• y = |x|: Parent function.
• |x + 2|: Shift left 2 units - New point at (-2, 0).
• |x - 3|: Shift right 3 units.

Square Root Functions

• y = √x: Parent function.
• -√x: Reflect over x-axis.
• √-x: Reflect over y-axis.
• Negative inside and outside: Reflect over origin.

Quadrant Analysis

• Positive x, Positive y: Quadrant 1.
• Positive x, Negative y: Quadrant 4.
• Negative x, Positive y: Quadrant 2.
• Negative x, Negative y: Quadrant 3.

Graphing Examples

• y = |x|, 2|x|, (1/2)|x|: Demonstrates vertical stretch and shrink.
• x² Transformation: Horizontal shift right 2, vertical shift up 3.

Detailed Graphing

Example: y = 3 - (x + 2)²

• Horizontal shift left 2 units.
• Vertical shift up 3.
• Parabola opens downward due to the negative sign.

Example: y = 4 - √(3 - x)

• Vertical shift up 4.
• Horizontal shift right 3 units.
• Reflects over origin (quadrant 3).

Parent Functions Overview

• y = x
• y = x²
• y = x³
• y = √x
• y = ∛x
• y = |x|

Key Takeaways

• Understand the effects of adding, subtracting, and multiplying functions.
• Distinguish between vertical and horizontal transformations by analyzing the function's formula.
• Practice graphing transformations to reinforce understanding.