Y-Axis
Geometry & MeasurementThe y-axis is the vertical number line on the coordinate plane, used to measure up and down position.
Definition
The y-axis is the vertical number line that goes up and down on the coordinate plane. Numbers above the center are positive; numbers below are negative.
Example
On the y-axis, $4$ is four spaces above the center and $-3$ is three spaces below. A point like $(0, 6)$ lies right on the y-axis.
Key Insight
The y-axis usually represents the output or result in a function. The y-intercept is where a line or curve crosses the y-axis, which is always the value when $x = 0$.
Definition
The y-axis is the vertical axis of the Cartesian coordinate system, the set of all points $(0, y)$ where $y$ is any real number. It is perpendicular to the x-axis and intersects it at the origin. Positive $y$ values are above; negative $y$ values are below.
Example
The y-intercept of a line $y = mx + b$ is found by setting $x = 0$, giving the point $(0, b)$. For $y = -3x + 7$, the y-intercept is $(0, 7)$. This is where the line crosses the y-axis.
Key Insight
For a function $f(x)$, the y-axis corresponds to the output values (range). A vertical line test uses the y-axis orientation: if any vertical line hits a graph more than once, the graph is not a function.
Definition
The y-axis is the second coordinate axis in $\mathbb{R}^2$, spanning the subspace $\{(0, y) : y \in \mathbb{R}\}$, the span of the standard basis vector $e_2 = (0, 1)$. Reflection across the y-axis is represented by the matrix $\begin{bmatrix}-1&0\\0&1\end{bmatrix}$. The y-axis divides the plane into the right and left half-planes.
Example
A function $f$ is even if $f(-x) = f(x)$ for all $x$, meaning its graph is symmetric about the y-axis. Cosine is even; its graph reflects identically across the y-axis. Even functions arise naturally in physics for symmetric potential wells.
Key Insight
In complex analysis, the imaginary axis plays the role of the y-axis when the complex plane is identified with $\mathbb{R}^2$ via $z = x + iy$. The symmetry between real and imaginary axes underlies deep results like the Schwarz reflection principle.