Point-Slope Form
Point-slope form is a way to write a linear equation using a known point on the line and the slope: y - y1 = m(x - x1).
Formula
y - y_1 = m(x - x_1)
Definition
Point-slope form writes the equation of a line using one known point $(x_1, y_1)$ on the line and its slope $m$: $y - y_1 = m(x - x_1)$. It is derived directly from the slope formula $m = (y - y_1)/(x - x_1)$ multiplied through, and is especially useful when two points are given, compute $m$ first, then apply the form. Geometrically it parameterizes the affine line through a fixed point with direction vector $(1, m)$ in $\mathbb{R}^2$, and it is the discrete analog of the tangent line approximation $f(x) \approx f(a) + f'(a)(x - a)$ used in differential calculus.
Example
A line with slope $3$ through $(2, 5)$ gives $y - 5 = 3(x - 2)$, which simplifies to $y = 3x - 1$. For the points $(1, -2)$ and $(4, 4)$, the slope is $m = (4-(-2))/(4-1) = 2$, so $y - (-2) = 2(x - 1)$ simplifies to $y = 2x - 4$. The tangent to $f(x) = x^2$ at $x = 3$ has slope $f'(3) = 6$, giving the tangent line $y - 9 = 6(x - 3)$, which is point-slope form with $(3, 9)$ and $m = 6$.
Key Insight
Think of it as saying "from this known point, the line keeps going at this slope": it is the algebraic version of the fact that a line is uniquely determined by a point and a direction, and either of two given points produces the same simplified equation. This is exactly the local-linearization idea from calculus: every differentiable function looks like its tangent line near a point, and that tangent is always written in point-slope form.