Foundations Teacher Notes Lesson 1 Guide • Functions Mastery
Foundations of Functions
Teacher Lesson Notes • Definitions, Variables, and Domain & Range
Algebra 1 Core Review Estimated Time: 45–50 min
Core Conceptual Definition
A relation is a function if and only if every input (\(x\)) has EXACTLY 1 output (\(y\)) .
Teacher Anchor Metaphor: Compare a function to a reliable soda vending machine. You press button B4 (one input), and a grape soda drops (one output). If pressing B4 sometimes gave grape soda and sometimes gave water, the machine is broken—not functional! However, buttons B4 and B5 can both dispense grape soda; multiple inputs may produce the same output.
Variables Breakdown
Independent Variable (\(x\)):
Functions on its own; chosen freely. The cause or input.
Examples: Hours worked, time elapsed, tickets sold.
Dependent Variable (\(y\)):
Happens as \(x\) changes; value depends on the input value.
Examples: Total pay earned, water temperature, total revenue.
Teaching Prompt: "Before graphing, always ask: Does the amount of money I earn depend on the hours I work, or do hours depend on money?"
Domain & Range
Domain:
The complete set of all \(x\)-values (input values).
\(\text{Domain} = \{x_1, x_2, x_3, \dots\}\)
Range:
The complete set of all \(y\)-values (output values).
\(\text{Range} = \{y_1, y_2, y_3, \dots\}\)
Teacher Tip (Set Notation): Write elements in ascending order without repeating duplicate values (e.g., if \(y=2\) appears twice, list it only once).
Teacher Worked Example • Discrete Set Analysis
Page 8 Direct Match
Task: Given relation \(S = \{(3,0), (4,2), (5,5), (1,2)\}\):
1. Identify Domain & Range:
• Domain (\(x\)-values): \(\{1, 3, 4, 5\}\)
• Range (\(y\)-values): \(\{0, 2, 5\}\) (2 appears twice, write once!)
2. Is it a function? Explain why:
YES! Every unique input (\(1, 3, 4, 5\)) pairs with exactly one output. There are no repeating \(x\)-values with differing \(y\)-values.
Algebra 1 Review Guide • Module 1: Functions Page 1 of 2
Lesson 1 Guide • Multi-Modal Representations
Testing Functions Across Representations
Page 8 Topic Coverage
Mapping Diagram
Rule: Each element in the domain has exactly one outgoing arrow .
X
1
3
4
5
→
Y
0
2
5
✓ 1 arrow leaves each X value
Table Representation
Rule: No repeating \(x\)-values assigned to different \(y\)-values.
✓ All x-values are unique
Vertical Line Test
Rule: Any vertical line crosses the graph at most 1 point .
✓ Passes Vertical Line Test
The Vertical Line Test (VLT) Deep Dive
Definition: A graph is a function if and only if every vertical line drawn on the coordinate plane intersects the graph at at most one point . If a vertical line touches the graph twice or more , that single \(x\)-coordinate corresponds to multiple outputs, which violates the function rule.
Anticipated Misconceptions & Exact Teacher Prompts
Misconception 1: "The y-value 2 repeats, so it can't be a function!"
Teacher Script: "Outputs can repeat! If you press Coke on two different vending buttons, it's fine. What you can NEVER have is pressing one button and getting two different drinks. Look ONLY at the \(x\)-column when determining function status."
Misconception 2: Confusing Vertical and Horizontal line tests.
Teacher Script: "To test if a graph is a function , slide a ruler vertically (up and down) across from left to right. We check if an \(x\)-value has multiple \(y\)-values. A horizontal line test tests for invertibility (one-to-one), not whether it is a function."
Quick Checks for Understanding (Cold Call Prompts)
"If relation \(A\) contains points \((2, 5)\) and \((2, -3)\), is it a function? Explain why." (No; input 2 yields two outputs)
"Can a circle drawn on a coordinate plane be a function of \(x\)? Why?" (No; vertical line passes through top and bottom)
"Why do we list \(\{0, 2, 5\}\) instead of \(\{0, 2, 5, 2\}\) when writing range?" (Sets only contain distinct elements)
Algebra 1 Review Guide • Module 1: Functions Page 2 of 2
Foundations Practice Worksheet Algebra 1 Practice
Function Foundations
Definitions, Variables, Representations, & Domain/Range
Name:
Date: Period:
1 Core Principles • Fill in the Blanks
Every (\(x\)) has EXACTLY 1 (\(y\)).
Domain: Set of all values.
Range: Set of all values.
2 Independent & Dependent Variables
Scenario A: Electric Vehicle Battery
The percentage of battery remaining in an electric car depends on the number of miles driven.
Independent Variable (\(x\)):
Dependent Variable (\(y\)):
Scenario B: Concession Stand Earnings
The total money earned by the booster club depends on the number of pretzel boxes sold.
Independent Variable (\(x\)):
Dependent Variable (\(y\)):
3 Multi-Representation Investigation
Given the relation: \(R = \{(3, 0), (4, 2), (5, 5), (1, 2)\}\)
Mapping Diagram
X
Y
Draw arrows from X to Y
Table of Values
x
y
Fill in ordered pairs
Coordinate Plot
Plot each point
Domain: { }
Range: { }
Is this relation a function?
Yes No
Why:
Functions Mastery • Lesson 1 Page 1 of 2
Lesson 1 • Identification Mastery
Vertical Line Test & Function Classification
Page 8 Topics
Vertical Line Test (VLT): A graph represents a function if and only if any vertical line drawn through the coordinate plane intersects the graph at no more than 1 point . If a vertical line touches in two or more spots, it is NOT a function.
4 Apply the Vertical Line Test to Each Graph
Graph A
[ ] Function [ ] Not a Function
Reason:
Graph B
[ ] Function [ ] Not a Function
Reason:
Graph C
[ ] Function [ ] Not a Function
Reason:
5 Determine if Each Representation is a Function
Item 1: Table Check inputs
Foundations Exit Ticket Exit Ticket 1 • Functions Mastery
Quick Check: Function Foundations
Name: Date:
1. The Core Rule:
Fill in the missing mathematical terms:
Every has EXACTLY one .
2. Function or Not?
\(P = \{(2, 5), (3, 8), (2, -1), (4, 8)\}\)
[ ] Function [ ] Not a Function
Why?
3. Domain and Range:
Using the set from Question 2:
Domain = { }
Range = { }
4. Vertical Line Test:
Sketch a quick graph that FAILS the VLT:
Score: ____ / 4
Cut Here
Exit Ticket 1 • Functions Mastery
Quick Check: Function Foundations
Name: Date:
1. The Core Rule:
Fill in the missing mathematical terms:
Every has EXACTLY one .
2. Function or Not?
\(P = \{(2, 5), (3, 8), (2, -1), (4, 8)\}\)
[ ] Function [ ] Not a Function
Why?
3. Domain and Range:
Using the set from Question 2:
Domain = { }
Range = { }
4. Vertical Line Test:
Sketch a quick graph that FAILS the VLT:
Score: ____ / 4
Evaluation Teacher Notes Lesson 2 Guide • Functions Mastery
Evaluating Functions
Teacher Lesson Notes • Numerical, Algebraic, & Expression Substitutions
Algebra 1 Core Review Estimated Time: 45–50 min
Instructional Meaning of Function Notation
Evaluating a function means finding the output value (\(y\)) when given a specific input value (\(x\)) .
Teacher Anchor Metaphor: Write \(f(x)\) on the board in giant letters. Emphasize: "The letter \(f\) is the name of the rule, and \(x\) is what goes into the machine. It is pronounced 'f of x', NEVER '\(f\) times \(x\)'. When you see \(f(6)\), it asks: 'What is the value of the function when \(x\) is replaced by 6?'"
Worked Example 1 • Direct Numerical Substitution
Page 8 Direct Match
Problem Statement:
\(f(x) = 4x + 3\). Find \(f(6)\).
\(f(6) = 4(6) + 3\)
\(f(6) = 24 + 3\)
\(f(6) = 27\)
Teacher Delivery Script:
"Replace the variable \(x\) with empty parentheses first: \(f(\; ) = 4(\; ) + 3\). Then drop 6 inside. Follow standard Order of Operations: multiply 4 by 6 first to get 24, then add 3. Remind students that the final output is 27, which corresponds to the coordinate point \((6, 27)\)!"
Worked Example 2 • Algebraic Expression Substitution
High Rigor • Page 8
Problem: Let \(g(x) = 2x^2 - 1\). Find the value of \(g(x + 1)\).
5-Step Solution:
\(g(x+1) = 2(x + 1)^2 - 1\)
\(g(x+1) = 2(x + 1)(x + 1) - 1\)
\(g(x+1) = 2(x^2 + 2x + 1) - 1\)
\(g(x+1) = 2x^2 + 4x + 2 - 1\)
\(g(x+1) = 2x^2 + 4x + 1\)
Instructional Safeguards:
Step 1: Replace \(x\) with the entire binomial \((x+1)\) inside parentheses.
Step 2 & 3: Warn students: \((x+1)^2 \neq x^2 + 1\)! They must expand using FOIL/box method: \((x+1)(x+1) = x^2 + 2x + 1\).
Step 4: Distribute the leading coefficient 2 across all three terms.
Step 5: Combine constant terms (\(+2 - 1 = +1\)).
Algebra 1 Review Guide • Module 2: Evaluation Page 1 of 2
Lesson 2 Guide • Graphical Evaluation
Evaluating Functions from Graphs
Page 8 Topics
Worked Example 3 • Reading Output from a Graph
Page 8 Direct Match
Graph of \(j(x)\)
Point: (3, 0) → \(j(3) = 0\)
The 3-Step Graph Reading Routine:
Step 1: Locate input on x-axis
Evaluation Practice Worksheet Algebra 1 Practice
Evaluating Functions
Numerical Inputs, Algebraic Expressions, & Coordinate Graphs
Name:
Date: Period:
1 Numerical Function Evaluation • Show All Substitution Steps
Problem 1: Given \(f(x) = 4x + 3\) Linear
Find the value of \(f(6)\):
Answer: \(f(6) = \)
Problem 2: Given \(f(x) = 4x + 3\) Linear
Find the value of \(f(-4)\):
Answer: \(f(-4) = \)
Problem 3: Given \(h(x) = x^2 - 3x + 5\) Quadratic
Find the value of \(h(5)\):
Answer: \(h(5) = \)
Problem 4: Given \(h(x) = x^2 - 3x + 5\) Quadratic
Find the value of \(h(-2)\):
Answer: \(h(-2) = \)
2 Algebraic Expression Substitution • Rigor Spotlight
Problem 5: Let \(g(x) = 2x^2 - 1\). Find and simplify the expression for \(g(x + 1)\).
Step 1: Replace \(x\)
Step 2: Expand \((x+1)^2\)
Step 3: Distribute 2
Step 4: Combine terms
Final Simplified Expression: \(g(x+1) = \)
Functions Mastery • Lesson 2 Page 1 of 2
Lesson 2 • Graphical Evaluation
Evaluating Functions from Graphs
Page 8 Topics
3 Analyze the Function Graph \(j(x)\)
Graph of \(j(x)\)
Each grid square represents 1 unit × 1 unit
Part A: Direct Input Evaluation
1. \(j(3) = \) (Point: (3, ___))
2. \(j(0) = \) (Point: (0, ___))
3. \(j(-2) = \) (Point: (-2, ___))
Part B: Reverse Evaluation (Find \(x\))
Find all values of \(x\) where \(j(x) = 0\):
\(x = \) (Hint: Look at the x-intercepts!)
4 Real-World Application • Function in Context
The height in feet of a dropped water balloon after \(t\) seconds is given by the function: \(h(t) = -16t^2 + 64\)
A. Evaluate \(h(1.5)\):
\(h(1.5) = \) ft
B. Interpret in Context:
What does your answer to Part A represent about the water balloon?
Functions Mastery • Lesson 2 Page 2 of 2
Evaluation Exit Ticket Exit Ticket 2 • Functions Mastery
Quick Check: Evaluating Functions
Name: Date:
1. Numerical:
\(f(x) = 5x - 7\)
Find \(f(4)\):
\(f(4) = \)
2. Expression:
\(g(x) = x^2 + 3\)
Find \(g(x + 2)\):
\(g(x+2) = \)
3. Graphical:
From the graph of \(j(x)\):
What is \(j(2)\)?
Score: ____ / 3
Cut Here
Exit Ticket 2 • Functions Mastery
Quick Check: Evaluating Functions
Name: Date:
1. Numerical:
\(f(x) = 5x - 7\)
Find \(f(4)\):
\(f(4) = \)
2. Expression:
\(g(x) = x^2 + 3\)
Find \(g(x + 2)\):
\(g(x+2) = \)
3. Graphical:
From the graph of \(j(x)\):
What is \(j(2)\)?
Score: ____ / 3
Graph Features Teacher Notes Lesson 3 Guide • Functions Mastery
Graph Features & Anatomy
Teacher Lesson Notes • Key Vocabulary, Extrema, & Intercepts
Algebra 1 Core Review Estimated Time: 45–50 min
Page 8 Vocabulary Glossary & Teaching Anchors
X-Intercept:
Where the graph crosses the \(x\)-axis. Also called zeros or roots . Always has coordinate \((x, 0)\).
Y-Intercept:
Where the graph crosses the \(y\)-axis. Starting value when input is 0. Always has coordinate \((0, y)\).
Line of Symmetry:
The vertical line \(x = h\) that cuts the graph directly in half, creating mirror-image sides.
Extrema:
General term for the extreme values of a graph: either a minimum or a maximum .
Minimum:
The lowest \(y\)-value of a graph (bottom of the valley / vertex).
Maximum:
The highest \(y\)-value of a graph (peak of the hill / vertex).
Visual Anatomy of a Parabolic Function
x-intercepts: \((-1, 0)\) and \((3, 0)\)
y-intercept: \((0, -0.75)\)
Minimum (Vertex): Lowest value is \(-1\) at \(x = 1\)
Line of Symmetry: Equation is \(x = 1\)
Teacher Alert • Value vs. Location of Extrema
Students frequently mix up the \(x\) and \(y\) coordinates when asked for the minimum or maximum. Teach this phrase explicitly: "The minimum VALUE is the \(y\)-coordinate (how low it goes). It OCCURS AT the \(x\)-coordinate (where it happens)."
Algebra 1 Review Guide • Module 3: Graph Anatomy Page 1 of 2
Lesson 3 Guide • Graph Intervals
Behavior Over Intervals
Page 8 Topics
Positive vs. Negative
Depends on location relative to the \(x\)-axis:
Positive Portion:
Where the graph is ABOVE the \(x\)-axis (\(y > 0\)).
Negative Portion:
Where the graph is BELOW the \(x\)-axis (\(y < 0\)).
Cut-off boundary points are the \(x\)-intercepts!
Increasing vs. Decreasing
Always read the graph from LEFT to RIGHT :
Increasing Portion:
Portions with positive slope (graph goes UP as you move left to right).
Graph Features Practice Worksheet Algebra 1 Practice
Graph Features & Vocabulary
Intercepts, Extrema, Symmetry, & Intervals
Name:
Date: Period:
1 Match Definitions to Key Terms
1. Where graph crosses the x-axis
2. Where graph crosses the y-axis
3. Line that cuts the graph in half
4. General term for max or min
5. Where graph is above the x-axis
6. Where graph is below the x-axis
2 Case Study • Complete Feature Breakdown of \(f(x)\)
Graph of \(f(x)\)
Grid scaled by 1 unit
x-intercept(s):
y-intercept:
Line of Symmetry: \(x = \)
Extrema Type: [ ] Min [ ] Max
Extrema Value:
Positive Portion: where \(x < \) & \(x > \)
Negative Portion: between \(x=\) & \(x=\)
Functions Mastery • Lesson 3 Page 1 of 2
Lesson 3 • Application
Interval Behavior & Sketching from Features
Page 8 Topics
3 Analyze Intervals of Increase & Decrease • Graph \(g(x)\)
Graph of \(g(x)\)
Peak: (-2, 2.5) Valley: (2, -2.5)
A. Relative Maximum:
The peak value is at \(x = \)
B. Relative Minimum:
The valley value is at \(x = \)
C. Interval of Decrease:
The curve falls between \(x = \) and \(x = \)
4 Graph Detective • Sketch from Given Clues
Construct a function graph satisfying:
x-intercepts at \((-3, 0)\) and \((3, 0)\)
y-intercept at \((0, 9)\)
Maximum value is \(9\)
Line of symmetry is \(x = 0\)
Positive on the interval \((-3, 3)\)
Your Coordinate Sketch
Functions Mastery • Lesson 3 Page 2 of 2
Graph Features Exit Ticket Exit Ticket 3 • Functions Mastery
Quick Check: Graph Features
Name: Date:
Graph of \(m(x)\)
Grid marks = 1 unit each
1. Intercepts:
x-intercepts:
y-intercept:
2. Extrema:
Type: [ ] Minimum [ ] Maximum
Extreme value:
3. Symmetry & Intervals:
Line of Symmetry: \(x = \)
From \(x=0\) to \(x=2\), graph is:
[ ] Increasing [ ] Decreasing
Score: ____ / 4
Cut Here
Exit Ticket 3 • Functions Mastery
Quick Check: Graph Features
Name: Date:
Graph of \(m(x)\)
Grid marks = 1 unit each
1. Intercepts:
x-intercepts:
y-intercept:
2. Extrema:
Type: [ ] Minimum [ ] Maximum
Extreme value:
3. Symmetry & Intervals:
Line of Symmetry: \(x = \)
From \(x=0\) to \(x=2\), graph is:
[ ] Increasing [ ] Decreasing
Score: ____ / 4
