Function Warmup Bell Ringer Warm-Up Algebra 1 • Unit 3
Function Foundations Bell Ringer
Activating Prior Knowledge: Inputs, Outputs, and Patterns
Name:
Date: Period:
Today's Goal: Explore how mathematical rules turn inputs into predictable outputs .
Time: 6–8 min
1 Task 1: The Function Machine
Follow the step-by-step rule
A number machine takes an input value (\(x\)) , applies a consistent rule, and produces an output value (\(y\)) .
Rule Inside Machine
Output = Multiply by 3, then add 2
Algebraic Rule: \(y = 3x + 2\)
Input (\(x\))
Work / Process
Output (\(y\))
Point \((x, y)\)
\(1\)
\(3(1) + 2\)
\(5\)
\((1, 5)\)
\(4\)
3( ) + 2
( , )
\(-2\)
3( ) + 2
( , )
2 Task 2: Pattern Sleuth • Inspecting the \(x\)-Values
Circle repeated inputs
Step 1: Circle every input value (\(x\)).Step 2: Check the box if any \(x\)-value repeats with different \(y\)-values.
Set A: Ordered Pairs
\(\{(-3, 4),\ (0, 7),\ (2, 4),\ (5, -1)\}\)
Inputs (\(x\)-values): ________________________________
Does any \(x\)-value repeat? Yes No
Set B: Ordered Pairs
\(\{(1, 8),\ (3, -2),\ (1, 5),\ (6, 0)\}\)
Inputs (\(x\)-values): ________________________________
Does any \(x\)-value repeat? Yes No
3 Task 3: Real-World Logic • The Vending Machine Paradox
Critical Thinking
The Button Test
Input: Press button B4
Expected Output: 1 Lemonade
Imagine pressing button B4 on Monday and getting a Lemonade. On Tuesday, you press B4 again, but it gives you a bottled water.
a) Would you say this vending machine is working properly and reliably? Why or why not?
b) Complete the sentence starter:
For a machine to be dependable, when I enter one specific input, I should get exactly ____________ (one / multiple) predictable output(s).
Ready for Notes: A function is a relation where each input has exactly one output! Bell Ringer • Page 1 of 1
Function Foundations Guided Notes Guided Notes Algebra 1 • Unit 3 • Lesson 1
Function Foundations & Representations
Essential Question: How do we determine if a relationship is a function, and how can we represent it?
Name:
Date: Period:
1 Core Concept: What is a Function?
Golden Rule
A relation is any set of ordered pairs \((x, y)\). A function is a special relation where:
Every INPUT (\(x\)) has EXACTLY ONE OUTPUT (\(y\)) .
The Inputs
\(x\)-values • Independent Variable
Domain = Set of all possible inputs
The Outputs
\(y\)-values • Dependent Variable
Range = Set of all possible outputs
2 Three Ways to Test: Is it a Function?
Method A: Mapping
Step 1: Look at the Input oval.Step 2: Count arrows leaving each number.Rule: If any input has more than 1 arrow , it is NOT a function.
\(x=2\) ⇒ 2 arrows \(y=5, 9\)
Result: NOT a Function
Method B: Table / Pairs
Step 1: Scan down the \(x\)-column.Step 2: Highlight repeating \(x\)'s.Rule: Same \(x\) paired with different \(y\)'s = NOT a function.
\((3, 7)\) and \((3, -1)\)
Repeating \(x\) with diff \(y\) = NOT
Method C: Vertical Line
Step 1: Imagine vertical lines.Step 2: Slide pencil across graph.Rule: If any line touches graph more than once , it FAILS the VLT.
Circles fail • Parabolas pass
Touches at most 1 pt = PASS
3 Guided Practice: Is Each Relation a Function? Justify! Teacher & Student Together
Example 1: Coordinate Set
\(\{(1, 4), (2, 5), (3, 5), (4, 8)\}\)
Step 1: List inputs:
Step 2: Does any input repeat with a different output?
Function? [ ] YES [ ] NO
Reason: Each input has exactly one output (even though \(y=5\) is shared!).
Example 2: Table
Step 1: Highlight repeated \(x\)'s: \(x = 5\)
Step 2: Are the \(y\)-values the same?
Function? [ ] YES [ ] NO
Reason: Input \(5\) has two outputs (\(12\) and \(20\)).
Example 3: Graph Inspection
Step 1: Draw vertical dashed line.
Step 2: Intersects graph at: 1 point
Function? [ ] YES [ ] NO
Reason: Passes the Vertical Line Test everywhere.
Continue to Page 2: Function Notation • Multiple Representations • Domain & Range Guided Notes • Page 1 of 2
Guided Notes • Part 2
Notation, Connecting Representations & Domain/Range
Algebra 1 • Unit 3
4 Function Notation: \(f(x)\)
“\(f(x)\)” means “\(f\) of \(x\)” (NOT multiplication!)
In function notation, \(f(x)\) is simply a fancy name for the output (\(y\)) . The letter inside parentheses is the input .
Case 1: Given Input, Find Output Evaluate
Problem: Given \(f(x) = 4x - 3\), find \(f(5)\).
Step 1 (Substitute): Replace \(x\) with \((5)\):
\(f(5) = 4(5) - 3\)
Step 2 (Calculate): Multiply then subtract:
\(f(5) = 20 - 3 = 17\)
Input is \(5\), Output is \(17\) → Point: \((5, 17)\)
Case 2: Given Output, Find Input Solve
Problem: Given \(g(x) = 2x + 6\), find \(x\) when \(g(x) = 20\).
Step 1 (Replace \(g(x)\)): Set equation equal to 20:
\(20 = 2x + 6\)
Step 2 (Solve for \(x\)): Subtract 6, divide by 2:
\(14 = 2x \implies x = 7\)
Input is \(7\), Output is \(20\) → Point: \((7, 20)\)
5 Connecting the 4 Representations
Verbal • Equation • Table • Graph
1. Verbal Context
A ride share costs \$4 base fee plus \$2 for every mile (\(m\)) driven.
Total Cost depends on miles driven.
2. Function Equation
Identify rate and start value:
\(C(m) = 2m + 4\)
\(m\) = miles, \(C(m)\) = cost (\$).
3. Table of Values
Each row forms \((m, C)\).
4. Coordinate Graph
Starts at \((0,4)\) with slope \(2\).
6 Domain and Range: The Step-by-Step Scan
Left → Right • Bottom → Top
Discrete Data (Individual points)
List individual numbers in braces \(\{ \dots \}\) from least to greatest:
Domain (\(x\)): \(\{-3, 0, 4, 7\}\)
Range (\(y\)): \(\{1, 5, 9\}\) (no repeats!)
Continuous Data (Connected line/curve)
Use inequality notation with minimum and maximum values:
Domain: \(\text{min } x \le x \le \text{max } x\)
Range: \(\text{min } y \le y \le \text{max } y\)
The “Bounding Box” Strategy Guided Scaffold
-3 4 3 -1
Scan Left → Right (\(x\)):
Scan Bottom → Top (\(y\)):
Domain: -3 \(\le x \le\) 4
Range: -1 \(\le y \le\) 3
Closed circle = \(\le\) or \(\ge\); Open circle = \(<\) or \(>\).
Next: Complete the Independent Practice to master each skill step by step! Guided Notes • Page 2 of 2
Function Practice Independent Work Practice Sheet Algebra 1 • Unit 3
Function Practice: Foundations & Notation
Step-by-step independent practice with guided prompts and frames
Name:
Date: Period:
1 Part 1: Function or Not? (Step-by-Step Analysis)
Test each input for exactly one output
1. Mapping Diagram
Domain (x) -3 1 4 Range (y) 2 9
Any input with >1 arrow? [ ] Yes [ ] No
Function? [ ] YES [ ] NO
Reason: Each input has ________________________________
2. Value Table
Repeating input: \(x =\) _________
Function? [ ] YES [ ] NO
Reason: Input \(4\) is paired with outputs _____ and _____.
3. Coordinate Pairs
\(\{(0, 3),\ (2, 3),\ (5, 3),\ (8, 3)\}\)
List Inputs (\(x\)): _____________________________
Function? [ ] YES [ ] NO
Reason: Each input has ________________________________
4. Graph Analysis
Vertical line touches: _____ point(s)
Function? [ ] YES [ ] NO
Reason: [Passes / Fails] Vertical Line Test because _________
2 Part 2: Function Notation Scaffolded Practice
Use the 3-step substitution method
5. Given \(f(x) = 5x - 7\), evaluate \(f(4)\).
Step 1: Insert parentheses: \(f(4) = 5(\phantom{xx}) - 7\)
Step 2: Multiply: \(f(4) = \phantom{xxx} - 7\)
Step 3: Final: \(f(4) =\)
Point \((x, y)\): \((4,\ \_\_\_\_)\)
6. Given \(g(x) = -3x + 10\), evaluate \(g(-2)\).
Step 1: Insert parentheses: \(g(-2) = -3(\phantom{xx}) + 10\)
Step 2: Note: negative × negative = positive!
Step 3: Final: \(g(-2) =\)
Point \((x, y)\): \((-2,\ \_\_\_\_)\)
7. Given \(h(x) = 4x + 9\), find \(x\) when \(h(x) = 25\).
Step 1: Set equal to 25: \(25 = 4x + 9\)
Step 2: Subtract 9: \(\phantom{xxxx} = 4x\)
Step 3: Divide by 4: \(x =\)
Point \((x, y)\): \((\_\_\_\_,\ 25)\)
8. Real-World Interpretation
\(H(t) = 12t + 40\) models the height \(H\) (in feet) of a balloon after \(t\) minutes.
a) Evaluate \(H(5)\):
b) Explain: “After 5 minutes, the balloon is __________________ feet high.”
Continue to Page 2 for Connecting Representations and Domain & Range Practice Sheet • Page 1 of 2
Practice Sheet • Part 3 & 4
Representations & Domain / Range Mastery
Algebra 1 • Unit 3
3 Part 3: The 4-Way Representation Link
Scenario → Equation → Table → Graph
The Scenario: Maya is saving for a new tablet. She starts with \$50 in her bank account and saves \$15 each week (\(w\)). Let \(S(w)\) represent her total savings.
A. Function Equation
Input (\(x\)): \(w\) (weeks)
Output (\(y\)): \(S(w)\) (total $)
Start = \$50 • Rate = +\$15/wk
Function Rule: \(S(w) = 15w + 50\)
B. Complete Value Table
Calculate weeks 2 and 4.
C. Coordinate Plane
0 30 60 90 120 0 1 2 3 4
Plot remaining points & connect!
4 Part 4: Domain & Range Determination
Discrete vs. Continuous
9. Discrete Relation (Listing in Braces)
Ordered pairs:
\(\{(-4, 2),\ (-1, 7),\ (3, 2),\ (6, -5)\}\)
Domain (\(x\)-values):
Range (\(y\)-values, no repeats):
Tip: List from least to greatest without repeats!
10. Continuous Graph (Inequality Notation)
(-3, -2) (3, 4)
Left-to-Right: min \(x\) → max \(x\)
Bottom-to-Top: min \(y\) → max \(y\)
Domain: \(\_\_\_\_\_\_ \le x \le \_\_\_\_\_\_\)
Range: \(\_\_\_\_\_\_ \le y \le \_\_\_\_\_\_\)
Solid circles = use \(\le\).
Error Analysis Challenge
Taylor claims: “The relation \(\{(1, 6), (2, 6), (3, 6)\}\) is NOT a function because the number \(6\) repeats three times.”
Is Taylor correct? [ ] Yes [ ] No Explain Taylor's mistake:
Ready for the Exit Ticket: Show your mastery of functions! Practice Sheet • Page 2 of 2
Function Check Exit Ticket Exit Ticket Algebra 1 • Unit 3 • Checkpoint
Function Foundations Mastery Check
Show what you know: Identification, Notation, Representations, and Domain/Range
Name:
Date: Period:
Target: Demonstrate 4 core skills with clear mathematical reasoning.
Score: / 4
1 Skill 1: Function Identification & Justification
1 Point
Relation Table
Is this a function? YES NO
Explain using the words input and output :
2 Skill 2: Function Notation
1 Point
Given the function: \(f(x) = 4x - 5\)
a) Evaluate \(f(-3)\):
Show substitution & calculation:
Answer: \(f(-3) =\)
b) Find \(x\) when \(f(x) = 23\):
Set up equation & solve for \(x\):
Answer: \(x =\)
3 Skill 3: Connecting Verbal & Algebraic Representations
1 Point
Scenario: A community pool charges a \$20 seasonal pass fee plus \$3 per visit (\(v\)). Let \(C(v)\) be total cost.
a) Write function equation:
b) Total cost for 8 visits:
Cost =
4 Skill 4: Domain and Range from a Graph
1 Point
(-2, -1) (4, 5)
Write inequality notation using the graph endpoints:
Domain (\(x\)): \(\le x \le\)
Range (\(y\)): \(\le y \le\)
Self-Reflection: Rate your confidence on today's concepts:
1: Need Help 2: Getting There 3: Got It 4: Can Teach It
Turn this ticket in to your teacher before leaving the classroom. Exit Ticket • Page 1 of 1
