Bring the class back together to discuss the challenge question. Have students complete a quick ticket out the door: "Give an example of a relation that is NOT a function using coordinate points."

Misconceptions Station

The "Repeating Outputs" Myth

Students often think a relation is not a function if outputs repeat (e.g., \( (1, 2) \) and \( (3, 2) \)).

➔ Fix: Reiterate that different passengers can stop at the same station. This is perfectly normal! Only repeating x's fails.

VLT Horizontal Confusion

Students mistakenly draw horizontal lines or perform horizontal scans instead of vertical scans.

➔ Fix: Have them slide an actual pen vertically across their page to show how the line represents \(x = a\).

Conductor's Callouts (Guiding Questions)

• "If I tell you the passenger name (input), is there any doubt which train (output) they are on?"
• "Why doesn't the vertical line test work for a circle? What input has multiple outputs?"
• "If a relation fails to be a function, does that mean it's useless? Or is it still a relation?"
• "How can we tell from a mapping diagram instantly if a relation is a function?"

Differentiation Routes

The Local Route (Support)

Provide physically cut-out mapping cards. Use yarn or string to connect input numbers to output numbers so they can touch and trace the paths manually.

The Express Route (Extension)

Challenge students to write equations that represent relations that are not functions (such as \(y^2 = x\) or \(x^2 + y^2 = 9\)) and explain why using graphs.

Function Junction Teacher Guide • Page 2 of 2 Metro Math Authority © 2026

y = x²

Does it pass the VLT? ____________

Explain: ___________________________________

Graph B: Perfect Circle

Does it pass the VLT? ____________

Explain: ___________________________________

5

Station 5: Final Dispatch

Conductor's Question:

Explain in your own words why a coordinate relation like { (2, 5), (2, 8), (3, 11) } is NOT a function.

Unit 2 • Function Junction Notes Page 2 of 2

PART D

Express Track Challenge

Construct Your Own Train Track

Draw a mapping diagram for a relation that contains at least four different points, but is NOT a function. Ensure you clearly label your Inputs (x) and Outputs (y).

Your Mapping Diagram

INPUTS (x)

➔

OUTPUTS (y)

Your Proof

List the coordinates of your custom relation:
Relation = { ______________________ }

Explain exactly why this relation is NOT a function:

Unit 2 • Practice Worksheet: Relations and Functions Page 2 of 2

Part B: Function Safety Inspections

3. Relation \(\mathcal{C}\) (Table)

Function? [ NO ]

Reason: Input 2 has two different outputs (5 and -4).

4. Relation \(\mathcal{D}\) (Set of Points)

Function? [ YES ]

Reason: Each input (1, 2, 3, 4) matches exactly one output (4).

Worksheet Page 2 Key

Part C: Graphical Rail Tests (VLT)

5. Continuous Curve (Parabola)

Function? [ YES ]

Reason: Passes the VLT; vertical lines cross the graph only once.

6. Closed Oval Shape (Ellipse)

Function? [ NO ]

Reason: Fails the VLT; vertical lines intersect the loop in two places.

Part D: Express Track Challenge

Sample Solution Criteria:

• Diagram: Student should map one input bubble number with two arrows pointing to different output bubble numbers.
• Coordinates: E.g., { (5, 10), (5, 11), (6, 12), (7, 13) }.
• Explanation: Must state that input 5 repeating with different outputs violates the rule of a function.

Function Junction Master Answer Keys • Page 2 of 3 Metro Math Authority © 2026

MASTER CONDUCTOR PASS

DEMONSTRATION OF LEARNING KEY

DOL Key

DOL 1: Mid-Route Junction Key

1. Routing Coordinates: { (-4, 3), (-2, 6), (0, 9), (2, 6) }

Domain = { -4, -2, 0, 2 }

Range = { 3, 6, 9 } (6 only listed once!)

2. Junction Safety Inspection

Function Choice: [ IT IS A FUNCTION ]

Justification: "Every input (\(x\)-value) is paired with exactly one output (\(y\)-value). No \(x\)-values repeat. Although the inputs -2 and 2 both map to the output 6, this is allowed as long as each unique input only maps to one partner."

DOL 2: Final Stop Clearance Key

1. Graphical Track Check (Sideways Curve)

Function Choice: [ NO ]

VLT Justification: "Fails the Vertical Line Test. If you draw a vertical line on the right side of the y-axis, it will cross the curve twice. This shows that a single input \(x\) has two different output \(y\) coordinates."

2. Real-World Transit Routing Scenario

Function Choice: YES, it is a function.

Justification: "Each individual destination has exactly one ticket price. If a single destination could cost two different random prices, it would not be a predictable, working system (which is what a function is)."

Function Junction Master Answer Keys • Page 3 of 3 Metro Math Authority © 2026

Equation Rescue Keys • Page 2 of 3 Metro Math Authority © 2026

MASTER DIAGNOSTIC CHECK

EQUATION QUIZ ANSWER KEY

Quiz Key

QUESTION 01 x = -7

\(-5x = 35\)

\(\frac{-5x}{-5} = \frac{35}{-5}\)
\(x = -7\)

⚠️ Misconception: Student divides by positive 5 or forgets division negative, writing \(x = 7\).

QUESTION 02 x = -16

\(\frac{x}{4} + 6 = 2\)

\(\frac{x}{4} + 6 - 6 = 2 - 6\)
\(\frac{x}{4} = -4\)
\(4 \cdot \left(\frac{x}{4}\right) = -4 \cdot 4\) ➔ \(x = -16\)

⚠️ Misconception: Subtracting incorrectly: \(2 - 6 = 4\), yielding \(x = 16\) instead of \(-16\).

QUESTION 03 x = 5

\(3x - 7 + 2x = 18\)

\(5x - 7 = 18\) (Combine 3x + 2x)
\(5x = 25\)
\(x = 5\)

⚠️ Misconception: Student tries to subtract \(2x\) from \(3x\) because it is on the other side of the constant, writing \(x - 7 = 18\).

QUESTION 04 x = 7

\(3(x - 2) = x + 8\)

\(3x - 6 = x + 8\) (Distribute)
\(2x - 6 = 8\)
\(2x = 14\) ➔ \(x = 7\)

⚠️ Misconception: Partial distribution: \(3x - 2 = x + 8\), yielding \(2x = 10 \rightarrow x = 5\).

Equation Rescue Keys • Page 3 of 3 Metro Math Authority © 2026