Relations, Functions, and Transformations

This comprehensive page covers the fundamental concepts of relations and functions, their properties, and various transformations. The content begins with basic definitions and progresses to more complex topics like graph transformations.

Definition: A relation is any set of (x,y) pairs, while a function is a special type of relation where each input has exactly one output.

Vocabulary: Domain refers to all possible input $x$ values, while Range encompasses all possible output $y$ values.

Example: Function notation uses f$x$ instead of y=... when representing a function.

Highlight: The vertical line test can be used to determine if a graph represents a function - if any vertical line intersects the graph more than once, it's not a function.

The page then delves into transformations of functions:

Definition: Translations shift the graph without changing its shape or orientation:

• Vertical shifts: f$x$ ± k (+ up, - down)
• Horizontal shifts: f$x + h$ (moves opposite of sign)

Highlight: For reflections:

• Across x-axis: -f$x$ maintains x-value but gives opposite y-value
• Across y-axis: f$-x$ gives opposite x-value but maintains y-value

Example: In dilations, when a > 1, the graph stretches vertically, and when 0 < a < 1, the graph compresses vertically.

The content concludes with notes on various representations of functions, including set mapping, ordered pairs, graphs, tables, words, and equations, providing a comprehensive foundation for understanding types of functions and equations.