Function Vending Machine & Graph Stories

A function is a promise: put in the same thing, get out the same thing — every single time. Test four suspicious vending machines, then learn to read a whole story from the shape of a graph.

Grade 8 · Unit 4 8.F.1 What a function is 8.F.2 Comparing functions 8.F.3 Linear vs. nonlinear 8.F.4 Building models 8.F.5 Graph stories

Part 1

The Vending Machine Floor

Four machines, one rule on trial: each input must produce exactly one output. Press inputs (repeat-press the same one — that's how you catch a cheater!), watch the log fill, and judge each machine after at least three presses.

🥤

Doubler +1

Insert a number. Something predictable might happen.

— insert an input below —

Press #	Input x	Output y

Part 2

Graph Stories

Read the story, then lock in the graph you predict before the reveal. Every rise, flat stretch, and fall in a graph is a sentence in the story.

Story 1 of 5

Part 3

The Function Machines

Three machines in the back room: build a function model from any two points, test any table for linearity, and settle any duel between two functions — even in different costumes.

Two points are all a linear function needs: slope first, then the starting value, then the equation.

Pick a job:

Linear functions add the SAME amount every step. Check the differences — they confess instantly.

Pick a function:

Two functions, different representations, one question at a time: whose rate is greater? Who starts higher? Who wins in the long run?

Pick a duel:

Part 4

Inspection Day Quiz

Ten questions to earn your machine-inspector's license. Wrong answers get a hint naming the exact machine or story that proves the right idea.

📦 Word Box

Function: A rule that assigns each input exactly ONE output. Same input in, same output out — every time.
Input (x): What goes into the machine — the independent variable.
Output (y): What comes out — the dependent variable. One per input, or the machine is disqualified.
Linear function: A function of the form y = mx + b; its graph is a straight line and its table adds the same amount every step.
Nonlinear function: Any function whose graph isn't a straight line — like y = x², whose differences keep growing.
Rate of change: How much the output changes per one unit of input — the slope m, a function's speed.
Initial value: The output when the input is 0 — the b in y = mx + b, where the story starts.
y-intercept: The point (0, b) where the graph crosses the y-axis — the picture of the initial value.
Increasing interval: A stretch where the graph rises left to right — the quantity is growing.
Decreasing interval: A stretch where the graph falls left to right — the quantity is shrinking.
Constant interval: A flat stretch — nothing changing, the story holding still.
Qualitative graph: A graph read for its shape and story — rises, falls, flats, curves — without exact numbers.