Y-Coordinate
The y-coordinate is the second number in an ordered pair, indicating a point's vertical distance and direction from the origin.
Definition
The y-coordinate (or ordinate) is the second number in an ordered pair $(x, y)$: it tells you how far to move up or down from the origin after moving left or right, positive meaning up and negative meaning down; for a function $y = f(x)$, the y-coordinate is the output for a given input $x$. Formally, it is the result of the projection $\pi_2 : \mathbb{R}^2 \to \mathbb{R}$ defined by $\pi_2(x, y) = y$, equivalent to the dot product with $e_2 = (0, 1)$; for a curve parameterized as $(x(t), y(t))$, the y-coordinate function $y(t)$ describes vertical position as a function of the parameter $t$.
Example
In the point $(3, 7)$, the y-coordinate is $7$, meaning move $7$ spaces up, while in $(5, -2)$ the y-coordinate $-2$ means move $2$ spaces down. For the parabola $y = x^2$, when $x = 3$ the y-coordinate is $9$, and when $x = -3$ it is also $9$, reflecting the even symmetry of the function about the y-axis. For the unit circle parameterized as $(\cos t, \sin t)$, the y-coordinate is $\sin t$, reaching its maximum of $1$ at $t = \pi/2$, the topmost point of the circle.
Key Insight
In slope-intercept form $y = mx + b$, the coefficient $m$ describes how the y-coordinate changes per unit increase in $x$ (the slope), and $b$ is the y-coordinate where the line crosses the y-axis. In differential geometry, the derivative $dy/dt$ gives the vertical component of the tangent vector of a curve, essential for computing arc length and curvature.