A function is a rule with one output for every input. A linear function is a starting amount plus a steady rate — b and m — and it can be told four ways: in words, in an equation, in a table and on a graph. Find the start, find the per, and say what they mean. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves — one of them on a coordinate plane you plot with a fingertip.
Start with where you already are before anything happens. Here that is 2.
b is where you start. m is how fast you go. y = mx + b. Every table, graph, story and equation in this unit is asking for those two numbers and what they mean.
Four things people get wrong
⚠️People often think…
A repeated number anywhere in the table breaks the function.
Repeated OUTPUTS are fine — lots of inputs can give the same answer. What breaks a function is one INPUT with two different outputs, because then the rule cannot tell you what to do.
Check the x column for repeats, not the y.
⚠️People often think…
Slope is run over rise.
Rise over run — the y's go on top. Flip it by accident and you get the reciprocal, which usually still looks like a reasonable answer, which is why this one is so expensive.
Rise first, because you climb before you walk away.
⚠️People often think…
In a table, the slope is just how much y jumps.
Only if x jumps by 1. Check the x steps first: if x goes up by 2 and y goes up by 6, the slope is 3, not 6.
Divide the y jump by the x jump. Every time.
⚠️People often think…
A vertical line has a slope of zero.
It is the other way around. A HORIZONTAL line has slope 0 — it rises nothing. A VERTICAL line is undefined, because the run is zero and you cannot divide by zero.
Flat: zero. Straight up: undefined.
Find b, find m, say what they mean
1Watch one
A gym charges $20 to join, then $5 a month. Write the rule.
The $20 happens once, at zero months.
That is b, the starting amount.
The $5 repeats every month — that is m.
y = 5x + 20. The word “per” always rides with x.
2Do one with me
A table: when x is 0, 2, 4, y is 7, 13, 19. Careful — x is not stepping by 1.
b is y when x is zero:
Each time, y jumps by
x jumps by 2, so m is
💬One sentence, then you move on
Why do you have to check the x steps before using the y jump?
3Try one
A perfectly flat, horizontal line. What is its slope?
I want a hint first
Walk along it. How far did you rise? Put that on top of how far you walked.
💬Last one — then you're done here
Why does one input with two different outputs break a function?
Where this goes
Where this lives
A paycheck with a base plus an hourly rate, a phone bill with a monthly fee plus data, a savings plan. Start plus rate, every time.
What this feeds
Next unit is transformations — slide, flip, turn and resize, where the whole shape moves by a rule.
Name one thing you pay for that has a starting fee and a rate.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a rule to a line — the ladder
Function (one output per input) → show it four ways (words, equation, table, graph) → linear means a constant change → slope m = rise ÷ run = change in y ÷ change in x → y-intercept b = y when x is 0 → y = mx + b → compare two functions by their m and their b.
Find the start. Find the per. Write y = (per)x + (start).
🤝 The whole topic in one idea
A linear function is a starting amount plus a steady rate. b is where you start; m is how fast you go. Every table, graph, equation and story in this unit is asking you to find those two numbers and say what they mean.
⚠️ Traps the test loves
A repeated OUTPUT is fine. A repeated INPUT with different outputs breaks a function.
Slope is rise over run — y’s on top. Upside down gives the reciprocal.
In a table, check the x steps. If x jumps by 2, divide the y jump by 2.
Swapping m and b: the number with “per” goes with x.
Horizontal line: slope 0. Vertical line: undefined. Not the other way around.
A distance-time graph is not a picture of a hill.
📐 Finding m and b in each form — the chart
Form
Slope m
y-intercept b
Equation y = mx + b
the number multiplying x
the number standing alone
Table
change in y ÷ change in x
the y in the row where x = 0
Graph
rise ÷ run between two clear points
where the line crosses the y-axis
Story
the amount per, each, every
the fee, the deposit, the amount at the start
🎯 How the test will ask
A table, a mapping or a graph — “is this a function?”
Two points — “what is the slope?” (subtract in the same order).
A story — “write the equation” and “what does the slope mean here?”
Two functions in two different forms — “which has the greater rate of change?”
Several equations or tables — “which is linear?”
A graph with no numbers — “describe what is happening.”
✅ Before the test, can you…
Decide whether a table or graph shows a function, and say why?
Find slope from two points, from a table and from a graph?
Find the y-intercept, even when x = 0 is not in the table?
Write y = mx + b from a story and say what m and b mean, with units?
Compare two functions given in different forms?
Tell linear from nonlinear in a table and in an equation?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: decide what is a function, name the two numbers that run every line, plot a table on a coordinate plane with your fingertip, then settle which plan is cheaper. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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