Function Mapping Guided Notes
Function Mappings & Arrow Notation
Algebra Foundations • Guided Notes
Name: ___________________
Date: _________
Period: _____
Part 1
What is Arrow Notation?
Input (Domain) → Output (Range)
Basic Mapping Notation
\( x \longrightarrow y \)
Read: "\(x\) maps to \(y\)"
Function Rule Notation
\( f: x \longmapsto 2x + 1 \)
Read: "\(f\) assigns \(x\) to \(2x+1\)"
Core Definition: In a function, each input value (\(x\)) maps to output value (\(y\)).
Part 2
Three Representations of the Same Mapping
Every relation can be shown in three formats. Model rule: "Add 3" (\(x \longrightarrow x + 3\)) for inputs \(\{1, 2, 3, 4\}\):
1. Mapping Diagram
Arrows connect input to output bubbles
Domain (x) 1 2 3 4 Range (y) 4 5 6 7
2. Input/Output Table
Organized rows of paired values
| Input (\(x\)) | Output (\(y\)) |
|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
| 4 | 7 |
3. Coordinate Pairs
Set of ordered pairs \((x, y)\)
\( \{ (1, 4), (2, 5), (3, 6), (4, 7) \} \)
• 1st value = Input (\(x\))
• 2nd value = Output (\(y\))
Part 3
Guided Conversion Practice
Example A: Translate mapping diagram into a table and coordinate pairs. \(x \longrightarrow 2x\)
2 3 5 4 6 10
| Input \(x\) | Output \(y\) |
|---|
| 2 | |
| 3 | |
| 5 | |
Set of Ordered Pairs:
\( \{ \) ( , ), ( , ), ( , ) \( \} \)
Example B: Given coordinate set \( S = \{ (-2, 4), (0, 0), (2, 4), (3, 9) \} \), draw the mapping arrows and fill table.
Draw arrows connecting pairs: -2 0 2 3 0 4 9
Notice: Outputs can receive multiple arrows! Both -2 and 2 map to: .
Function Mappings & Arrow Notation Page 1 of 2
The Arrow Test: Function vs. Not a Function
Analysis & Mastery
The Golden Rule of Arrow Diagrams
Count the arrows leaving each input!
IT IS A FUNCTION IF:
Every input has exactly one arrow pointing away from it. Inputs are allowed to point to the same target output.
IT IS NOT A FUNCTION IF:
Any input has two or more arrows branching out from it (one single input produces conflicting outputs).
Case 1: One-to-One A B 1 2 FUNCTION
1 arrow leaves each input
Case 2: Many-to-One A B 1 FUNCTION
Outputs can share targets!
Case 3: One-to-Many A 1 2 NOT A FUNCTION
Input A has 2 outgoing arrows!
Part 4
Determine: Function or Not? Justify Your Answer.
1 2 3 8 9
Relation 1: [ ] Function [ ] Not a Function
Reason: Every input has exactly arrow leaving it.
4 7 1 3 5
Relation 2: [ ] Function [ ] Not a Function
Reason: Input maps to multiple outputs: .
Part 5
Independent Application: The Vending Machine
Context: A vending machine registers button presses (input \(x\)) and dispenses drinks (output \(y\)):
| Button Pressed (\(x\)) | Item Dispensed (\(y\)) |
|---|
| B1 | Water |
| B2 | Juice |
| B3 | Water |
| B4 | Soda |
Draw arrows connecting inputs to outputs: B1 B2 B3 B4 Water Juice Soda
Is this machine a function?
[ ] YES [ ] NO
Explain with arrow rule:
Quick Recall: Arrows leave the Domain (\(x\)) & point to the Range (\(y\)). 1 arrow per input = Function!
Page 2 of 2
Translation Notation Guided Notes
Translations & Arrow Notation
Geometric Rigid Motions • Guided Notes
Name: ___________________
Date: _________
Period: _____
Part 1
What is a Translation?
Rigid Motion (Isometry)
A translation is a transformation that (moves) every point of a figure the exact same in the exact same .
Pre-Image The original figure \( A(x, y) \)
Arrow Motion The slide vector rule \( \longrightarrow \)
Image The transformed figure \( A'(x', y') \) (A prime)
Part 2
Decoding Coordinate Arrow Notation
\( (x, y) \longrightarrow (x + a, \; y + b) \)
"The original point \((x, y)\) maps to the new point \((x + a, y + b)\)"
Horizontal Shift: Affects \(x\)
- \( (x + a) \implies \) Shift RIGHT + positive
- \( (x - a) \implies \) Shift LEFT - negative
Vertical Shift: Affects \(y\)
- \( (y + b) \implies \) Shift UP + positive
- \( (y - b) \implies \) Shift DOWN - negative
Part 3
Matching Verbal Descriptions & Arrow Rules
| Verbal Description | Coordinate Arrow Notation |
|---|
| Slide 4 units right and 3 units down | \( (x, y) \longrightarrow (x + 4, y - 3) \) |
| Slide 6 units left and 5 units up | \( (x, y) \longrightarrow ( \) , \( ) \) |
| Slide 2 units right and 0 units vertically | \( (x, y) \longrightarrow ( \) , \( ) \) |
| write description | \( (x, y) \longrightarrow (x - 7, y - 1) \) |
Part 4
Model Walkthrough: Translating \(\triangle ABC\)
x y A B C A' B' C' Rule: \( (x, y) \to (x + 3, y + 1) \)
Apply Rule: \( (x, y) \longrightarrow (x + 3, y + 1) \)
| Pre-Image Point | Substitute & Calculate | Image Point |
|---|
| \( A(-2, 2) \) | \( (-2 + 3, \; 2 + 1) \) | \( A'(1, 3) \) |
| \( B(-2, -1) \) | \( (-2 + 3, \; -1 + 1) \) | \( B'(1, 0) \) |
| \( C(0, -1) \) |
