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Algebra · Straight-line graphs

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Gradient and intercept

Why gradient is rise over run (in that order, signs included), what m and c each control in y = mx + c, and how to test whether a point lies on a line.

Algebra · Straight-line graphs

Gradient and intercept

Why gradient is rise over run (in that order, signs included), what m and c each control in y = mx + c, and how to test whether a point lies on a line.

Why it works

A straight line has one defining property: it climbs at a constant rate. The gradient measures that rate —

m=riserun=y2−y1x2−x1m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}

— how much yy changes for each step of 1 in xx. Through (1,2)(1, 2) and (4,11)(4, 11): rise =9= 9, run =3= 3, so m=3m = 3 — each unit right climbs 3 up.

Order and signs are the whole game. Subtract the coordinates in the same order top and bottom. Through (−2,7)(-2, 7) and (4,−5)(4, -5): m=−5−74−(−2)=−126=−2m = \frac{-5 - 7}{4 - (-2)} = \frac{-12}{6} = -2 — a downhill line has a NEGATIVE gradient, and mixing the order (−5−7−2−4\frac{-5-7}{-2-4}) silently flips the sign. Run-over-rise (xy\frac{x}{y}) is the other classic inversion; the word order "rise over run" is the antidote.

In y=mx+cy = mx + c, the two constants have jobs. mm is the gradient; cc is the yy-intercept — where the line crosses the yy-axis (because x=0x = 0 there, leaving y=cy = c). Read carefully when the form is shuffled: y=5−2xy = 5 - 2x has gradient −2-2 and intercept 55 — the gradient is the number GLUED TO xx, wherever it sits.

A point lies on a line exactly when it satisfies the equation. Does (5,17)(5, 17) lie on y=3x+2y = 3x + 2? Substitute: 3(5)+2=173(5) + 2 = 17 ✓ — yes. No graph needed; the equation IS the line, listed as coordinates.

Crossing the axes. The line meets the yy-axis at x=0x = 0 (giving cc) and the xx-axis at y=0y = 0 — solve 0=mx+c0 = mx + c. For y=3x+2y = 3x + 2: x=−23x = -\frac{2}{3}.

Unknown coordinates ride along. If the line through (2,k)(2, k) and (5,13)(5, 13) has gradient 4, then 13−k3=4\frac{13 - k}{3} = 4, so 13−k=1213 - k = 12 and k=1k = 1 — the gradient formula becomes an equation in kk.