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Point-slope form is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is any known point on the line. This form is most useful when you know the slope and a point (or two points, from which you compute the slope). It comes directly from the slope definition: m = (y − y₁)/(x − x₁), rearranged. Point-slope form is often the fastest way to write a linear equation, and it converts easily to slope-intercept form by distributing and solving for y. It also appears in calculus as the basis for linear approximation.
Derive it from the slope formula so students see it is not an arbitrary form but a rearrangement. Practice writing equations given a slope and a point, then given two points (find slope first, then use either point). Convert to slope-intercept form and verify both forms produce the same graph. Emphasize that either point can serve as (x₁, y₁) — the result is the same line.
You already know that slope measures the steepness of a line — the ratio of vertical change to horizontal change, rise over run. Point-slope form isn't a new idea layered on top of that; it's just the slope definition written in a slightly rearranged way. Starting from slope = (y − y₁)/(x − x₁), multiply both sides by (x − x₁) and you get y − y₁ = m(x − x₁). That's the whole derivation. The form is worth naming because it's often the fastest route from information to equation.
The key insight is what information the form demands. To write an equation for a line, you need two pieces of data: the slope and one specific location. Point-slope form accepts exactly that. If you're given slope m = 3 and the point (2, 5), write immediately: y − 5 = 3(x − 2). No intermediate steps required. If you're given two points instead, compute the slope first (m = (y₂ − y₁)/(x₂ − x₁)), then use either point in the form — you'll get the same line either way.
Watch the signs carefully. The form is y − y₁, so if your point has a negative coordinate, subtraction of a negative becomes addition. For the point (−3, 4) with slope 2: y − 4 = 2(x − (−3)), which simplifies to y − 4 = 2(x + 3). A common error is to write x − 3 here, dropping the double negative. Reading the formula literally — "y minus y₁" and "x minus x₁" — and substituting the actual values of y₁ and x₁ prevents this mistake.
To convert to slope-intercept form, simply distribute m and solve for y. From y − 4 = 2(x + 3): distribute to get y − 4 = 2x + 6, then add 4 to both sides: y = 2x + 10. You'll also encounter point-slope form again in calculus, where the tangent line to a curve at a point (a, f(a)) has slope f′(a). The tangent line equation is y − f(a) = f′(a)(x − a) — exactly point-slope form, making this the algebraic template for local linear approximation.