Point-Slope Form
AlgebraPoint-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 is a way to write the equation of a line when you know one point on the line and the slope. The formula is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the known point and $m$ is the slope.
Example
A line with slope $3$ passes through the point $(2, 5)$. Plug in: $y - 5 = 3(x - 2)$. Simplify: $y = 3x - 1$.
Key Insight
Point-slope form is a great starting tool when you have a specific point. Think of it as saying "from this known point, the line keeps going at this slope."
Definition
Point-slope form $y - y_1 = m(x - x_1)$ defines a line through point $(x_1, y_1)$ with slope $m$. It is derived from the slope formula: $m = (y - y_1)/(x - x_1)$, then multiplied through. It is especially useful for writing an equation when two points are given (compute $m$ first, then apply the form).
Example
Find the equation through $(1, -2)$ and $(4, 4)$. Slope: $m = (4-(-2))/(4-1) = 6/3 = 2$. Point-slope: $y - (-2) = 2(x - 1)$, so $y + 2 = 2x - 2$, giving $y = 2x - 4$.
Key Insight
Point-slope form is the algebraic version of the geometric fact that a line is uniquely determined by a point and a direction (slope). Either point can be used and the result simplifies to the same line equation.
Definition
Point-slope form is a parameterization of the affine line through a fixed point $(x_1, y_1)$ with direction vector $(1, m)$ in $\mathbb{R}^2$. It can be derived from the defining condition that slope is constant: $(y - y_1)/(x - x_1) = m$ for all $x$ not equal to $x_1$. This form is the discrete analog of the tangent line approximation $f(x) \approx f(a) + f'(a)(x - a)$ used in differential calculus.
Example
The tangent to $f(x) = x^2$ at $x = 3$ has slope $f'(3) = 6$, giving tangent line $y - 9 = 6(x - 3)$, or $y = 6x - 9$, which is point-slope form with $(3, 9)$ and $m = 6$.
Key Insight
Point-slope form is precisely the local linearization formula from calculus. Every differentiable function looks like its tangent line near a point, and the tangent is always written in point-slope form.