Linear Functions

A taxi meter starts at 3 dollars before the wheels even turn — the flag fall — and then adds 2.50 dollars for every kilometre you ride. A phone plan bills 15 dollars a month plus 0.10 per minute of calls. A plumber charges a 60-dollar call-out fee plus 45 an hour. Different jobs, same skeleton: a starting amount, plus a fixed rate times how much you use.

Every one of these is a linear function — the functions whose graph is a perfectly straight line. And they are special for one reason that the rest of calculus will keep circling back to: a linear function is the only kind whose rate never changes. The taxi's tenth kilometre costs exactly what its first did. Nothing speeds up, nothing tapers off. Every linear function fits one tidy template:

y = mx + b

This is called slope-intercept form. Just two numbers, m and b, completely describe the line — change either one and the line tilts or slides. Better still, each number is something you can point at in the real situation: m is the rate (dollars per kilometre, cost per minute), and b is the starting amount (the flag fall, the monthly fee). For the taxi, y = 2.5x + 3 — and you already know what both numbers mean. (Some books write y = mx + c; same idea, different letter for the starting value.)

Every linear function can be written as f(x) = mx + b for two constants m and b.
m is the slope: the constant rate of change. Increasing x by 1 always changes y by exactly m — the same everywhere on the line.
b is the y-intercept: the starting value, b = f(0), where the line crosses the y-axis.

b — where the line starts

The number b is the y-intercept: the height where the line crosses the y-axis. That makes sense — at the crossing point x = 0, so

y = m(0) + b = b.

So the line always passes through the point (0, b). In a real tariff, x = 0 means "before anything has happened yet" — zero kilometres ridden, zero minutes called — so b is the amount you owe just for showing up: the flag fall, the standing charge, the joining fee. Increasing b lifts the whole line up; decreasing it drops the line down, without changing its tilt at all.

m — the rate that never changes

The number m is the slope: it sets how steeply the line rises or falls, because it is the amount y changes every single time x steps up by 1. Read it as a "per": dollars per kilometre, litres per minute, metres per second. A bigger m tilts the line up more sharply; a negative m tips it downhill (a phone battery draining, a bath emptying); m = 0 leaves it perfectly flat — a quantity that never changes at all. (We'll measure slope precisely in the next lesson.)

Drive the two sliders below and watch how m and b shape the line. The dot marks the y-intercept (0, b). Notice the two controls never interfere: b slides the line up and down rigidly, while m pivots it around that dot — the intercept is the hinge.

Worked example: from a tariff to an equation — and back

Forwards. A taxi charges 3 dollars flag fall, then 2.50 dollars per kilometre. Build the fare function for a trip of x kilometres:

the starting amount is the flag fall, so b = 3;
the rate is per-kilometre cost, so m = 2.5;
therefore y = 2.5x + 3.

Now the function answers questions for free. An 8-kilometre trip costs y = 2.5(8) + 3 = 20 + 3 = 23 dollars. A 20-kilometre airport run: 2.5(20) + 3 = 53. One formula, every fare.

Backwards. Your phone bill is computed by y = 0.10x + 15, with x in minutes. Read the two facts straight off: the 15 sits where b lives, so it's the standing charge — you pay it even if you never call anyone (x = 0 gives y = 15). The 0.10 multiplies x, so it's the rate: each extra minute costs ten cents, the first minute exactly as much as the thousandth. Translating both ways — story \to equation and equation \to story — is the whole skill.

Worked example: the line through two points

Often you aren't handed m and b — you're handed two data points and asked for the line through them. Say a courier charges 5 dollars for a 1-kg parcel and 13 dollars for a 5-kg one… but let's do the sums with the points (1, 3) and (5, 11). The recipe is always slope first, then intercept.

Step 1 — slope. Between the points, x ran from 1 to 5 (a run of 4) while y rose from 3 to 11 (a rise of 8). The rate is rise per unit of run:

m = \frac{\Delta y}{\Delta x} = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2.

Step 2 — intercept. Now that m = 2 is known, feed either point into y = 2x + b and solve. Using (1, 3): 3 = 2(1) + b, so b = 1. The line is y = 2x + 1.

Step 3 — check. The other point must also fit: 2(5) + 1 = 11. ✓ Step through the construction:

This two-step recipe never changes: m from \Delta y / \Delta x, then b from one point. Two points pin down a line completely — which is exactly why two numbers (m and b) suffice to describe it.

Three snags catch nearly everyone with lines:

Slope is rise over RUN — \Delta y / \Delta x, never \Delta x / \Delta y. The change in y goes on top. Flip the fraction and a gentle slope of \tfrac{1}{4} masquerades as a steep 4. A memory hook: slope answers "how much does y change per one step of x?" — the per-thing always goes underneath, just like "dollars per kilometre".
A negative slope falls left-to-right. Read a graph the way you read a sentence: left to right. m < 0 means each step right takes you down. A line like y = -2x + 7 is still perfectly linear — its rate is constant, it just happens to be a constant loss.
"Linear" has a stricter cousin. In school maths, any y = mx + b is called linear. But to a mathematician a truly linear map must satisfy f(2x) = 2f(x) — double the input, double the output — which only works when b = 0. (The taxi fails it: an 8-km ride costs 23, not double the 13 of a 4-km ride, because the flag fall doesn't double.) Lines with b \neq 0 are technically called affine. Nobody will dock you marks for saying "linear" — but when you meet linear algebra later, this fine print suddenly matters.

Drawing a line from just two numbers

Given an equation in slope-intercept form, you can draw its graph with no table of values at all — the two numbers are the drawing instructions. The recipe: start at b, then walk the slope. Put your pencil on (0, b); then, from that point, go 1 right and m up (down if m is negative), mark the new point, and rule the line through them.

Take y = 3x - 2: the slope is m = 3 and the y-intercept is b = -2. So: pencil on (0, -2); walk right 1, up 3, arriving at (1, 1); join and extend. Step through the worked line below and watch the run-and-rise staircase do the walking.

Khan Academy introduces slope-intercept form here:

Almost nothing in nature is actually linear. Populations grow in curves, planets sweep ellipses, your phone battery drains faster when it's cold. So why does every science course, every economics model, every engineering estimate reach for y = mx + b first?

Because of a marvellous cheat: zoom in far enough on any smooth curve, and it looks like a straight line. The Earth is a sphere, but your street looks flat — you are living on a zoomed-in patch of a curve. A curve's zoomed-in stand-in line is called its tangent line, and near the point of contact it's a superb approximation: simple to compute with, easy to reason about, wrong by only a whisker. That's why linear models rule the world — not because the world is linear, but because up close, everything behaves as if it were.

Here's the punchline for this course: calculus is the machine for finding those lines. The derivative you're heading toward is exactly "the slope m of the line a curve pretends to be at a point." Master m and b now, and you've already learned the alphabet the whole rest of calculus is written in.