← all units · Unit 03 · Functions
Linear functions
If something changes at a steady rate — your pay, a phone plan, miles down the road — it's a line. And a line is just two numbers: how fast it changes and where it starts.
Where you'll actually use this
Hourly pay is a line: total = rate × hours + any base. Overtime, commissions, and salaries are all bends on this one idea.
Phone, electric, streaming, the gym — a monthly fee plus a per-use rate. Comparing two plans is just comparing two lines.
The intercept is where you begin before anything happens — the flat fee, the base pay, the gas already in the tank.
The idea
A line is the shape of steady change. The equation packs the whole story into two numbers — a slope and a starting value.
y = m x + b
- m
- slope — the rate of change ("rise over run")
- b
- y-intercept — the starting value (y when x = 0)
- x
- the input (hours, months, miles…)
- y
- the output (pay, cost, distance…)
Slope + climbs, − falls, 0 is flat. Steeper number = faster change.
Worked example A gym charges $25 to join plus $15/month. Cost after 6 months: y = 15(6) + 25 = 90 + 25 = $115. The $25 is where the line starts; the $15 is how fast it climbs.
Your turn
Three kinds of question, fresh numbers every time. Miss one and the full worked solution appears — seeing the steps is the point.
Loading a problem…
Match the Line
Dial the slope and the intercept until your line lands on the dashed target. It's the fastest way to feel what m and b actually do.
Land your line on the dashed target.
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Next in this unit: reading a line off a graph, then parallel & perpendicular. Feeds straight into systems of equations (two lines, one crossing point).