L2 · Slope from two points

Learning goal

“I can calculate the slope of a line from any two points on it, using m = rise/run = (y2 - y1)/(x2 - x1), and I can read a slope's sign and size: positive climbs, negative falls, zero is flat, and a vertical line's slope is undefined.”

Slope (m) — the steepness of a line, with magnitude and direction; the rate of change of the function.

Rise — the change in y between two points: y2 - y1.

Run — the change in x between the same two points, in the same order: x2 - x1.

Undefined slope — what a vertical line has: its run is 0, and division by 0 is impossible, so no slope number exists.

Worked example

Find the slope of the line through (1, 2) and (4, 8). Step 1: Rise = change in y = 8 - 2 = 6. Step 2: Run = change in x, in the same order = 4 - 1 = 3. Step 3: Slope m = rise/run = 6/3 = 2. Step 4: Check by reordering: using (4, 8) first, rise = 2 - 8 = -6 and run = 1 - 4 = -3, and (-6)/(-3) = 2 - the same slope. The order of the points does not matter; what matters is that rise and run are subtracted in the same order. Say it plainly: slope 2 means "up 2 for every 1 across" - it is the constant difference from L1, now measured between any two points, not just neighbours.

Check yourself — explain it

Reversing the order of the two points gives the same slope because...

Paired problem — example still visible

Find the slope through (0, 3) and (2, 9). -> Rise 9 - 3 = 6, run 2 - 0 = 2, slope 6/2 = 3.

Faded completion — last step is yours

"A line with slope 2 passes through (2, 5) and (6, ___). The missing y is" -> 13 (run 4, so rise 8; 5 + 8 = 13)

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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