y = mx + b: reading a line

Two numbers pin down any non-vertical line: where it starts, and how it moves. The form y = mx + b puts both on display — which is why it's the most useful way to write a line.

m and b, the two controls

• b is the y-intercept: the line's height when x = 0, the point (0, b) where it crosses the y-axis. Change b and the whole line slides up or down.
• m is the slope from last lesson. Change m and the line tilts around its intercept.

They're independent: one sets the starting height, the other sets the rate.

Reading a line in one glance

y = −2x + 5 says: start at (0, 5), and fall 2 for every 1 step right. That's the whole graph, spoken. A phone plan of 30 dollars flat plus 4 per gigabyte is the same sentence: cost = 4g + 30 — slope 4, intercept 30. b is the starting value; m is the rate of change.

Graphing in seconds

1. Dot the intercept (0, b).
2. From it, step the slope: for m = 3/4, go right 4, up 3. Dot again.
3. Connect. Done — two points determine the line.

Building the equation from data

Given slope 2 through the point (3, 1): start with y = 2x + b, feed the point in — 1 = 2(3) + b — and solve: b = −5. So y = 2x − 5. Given two points instead, compute m first, then reuse this move with either point.

Why this matters

y = mx + b is the workhorse form for everything next: systems (two of these, crossing), inequalities (one of these, shading a side), and regression in statistics (the best-fitting one of these through a data cloud). If you can read m and b, you can read them all.