X-Axis
Geometry & MeasurementThe x-axis is the horizontal number line on the coordinate plane, used to measure left and right position.
Definition
The x-axis is the horizontal number line that goes left and right across the coordinate plane. Numbers to the right of the center are positive; numbers to the left are negative.
Example
On the x-axis, the number $3$ is three spaces to the right of center, and $-2$ is two spaces to the left. When you plot a point like $(5, 0)$, it sits right on the x-axis.
Key Insight
The x-axis represents the starting position before you go up or down. Any point that lies on the x-axis has a y-coordinate of $0$.
Definition
The x-axis is the horizontal axis of the Cartesian coordinate system, defined by the set of all points $(x, 0)$ where $x$ is any real number. It intersects the y-axis at the origin $(0, 0)$. Positive $x$ values are to the right; negative $x$ values are to the left.
Example
The x-intercept of a line is where it crosses the x-axis (where $y = 0$). For $y = 2x - 6$, set $y = 0$: $0 = 2x - 6$, $x = 3$. The x-intercept is $(3, 0)$.
Key Insight
In function notation, the x-axis holds the input values (domain) of a function. The x-intercepts of a function are also called its zeros or roots, the values where the function equals zero.
Definition
The x-axis is the first coordinate axis in $\mathbb{R}^2$, spanning the subspace $\{(x, 0) : x \in \mathbb{R}\}$. In linear algebra terms, it is the span of the standard basis vector $e_1 = (1, 0)$. A reflection across the x-axis is represented by the matrix $\text{diag}(1, -1)$.
Example
The x-axis is the set of fixed points of the reflection matrix $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$. Eigenvalues of this matrix are $+1$ (x-axis, eigenvector $(1,0)$) and $-1$ (y-axis, eigenvector $(0,1)$).
Key Insight
The choice of axes is a convention, not a geometric necessity. Rotating the coordinate system by angle $\theta$ replaces the axes with new orthogonal directions, illustrating that the x-axis is a human-chosen reference direction within $\mathbb{R}^2$.