Y-Coordinate
Geometry & MeasurementThe 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 is the second number in an ordered pair $(x, y)$. It tells you how far to move up or down from the origin after you have moved left or right. Positive means up; negative means down.
Example
In the point $(3, 7)$, the y-coordinate is $7$, meaning move $7$ spaces up. In the point $(5, -2)$, the y-coordinate is $-2$, meaning move $2$ spaces down.
Key Insight
The y-coordinate is sometimes called the "ordinate." For any function, the y-coordinate is the output value. The y-intercept of a graph is always the y-coordinate when $x = 0$.
Definition
The y-coordinate (ordinate) of a point $(x, y)$ is its vertical displacement from the origin: positive values are above the x-axis, negative values below. For a function $y = f(x)$, the y-coordinate is the output for a given input $x$.
Example
For the parabola $y = x^2$, when $x = 3$ the y-coordinate is $9$; when $x = -3$ the y-coordinate is also $9$. The symmetric y-values reflect the even symmetry of the function about the y-axis.
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.
Definition
The y-coordinate 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
For the unit circle parameterized as $(\cos t, \sin t)$, the y-coordinate is $\sin t$. The maximum y-coordinate of $1$ occurs at $t = \pi/2$ (the topmost point), illustrating how the y-coordinate function encodes the geometric height of the curve.
Key Insight
In differential geometry, the y-coordinate of a curve at a given $t$ is the "second component of the position vector." The derivative $dy/dt$ gives the vertical component of the tangent vector, essential for computing arc length and curvature.