In arithmetic, the vertical axis is a part of a two-dimensional graph of points. It defines one of the two dimensions. It is demarcated so that a point's height can be measured against it. The convention is that a point (P) with coordinates (x,y) has a height (y). For example, a point (5,4) will be level with the fourth demarcation on the vertical axis.
The Vertical Axis in the Cartesian Plane
The vertical axis is part of what is called the "Cartesian plane," or what is known more colloquially as the x-y plane. The vertical axis intersects the horizontal axis at what is called the "origin." This is the point (0,0). The Cartesian plane is parametrized with coordinates, which means that the vertical axis is broken up into units. A point (x,y) on the graph will have one value corresponding to its height and one corresponding to its lateral position. For example, the projection of the point (1,2) onto the vertical axis is 2.
Representation of the Vertical Axis
The vertical axis is represented as a vertical line with arrows at the top and bottom to indicate that it continues indefinitely in both the positive and negative direction. The vertical axis is therefore a copy of the real number line, just made vertical.
Use of the Vertical Axis for Functions
The vertical axis is reserved for a function. A function is a curve that produces only one output value for each input value x. The function is usually represented with either the letter "y" or by f(x). The latter notation indicates that the function's value depends on the value of x, i.e., the corresponding point on the horizontal axis. The horizontal axis is therefore reserved for the variable that the function is a function of.
One Vertical Axis Value Per Input
By choosing the vertical axis to be the axis for the function values, it holds that any vertical line drawn on the graph will pass through the function's curve no more than once. So for each value for the horizontal axis, the vertical axis has only one value. This is a desirable quality akin to getting only one answer to any question you ask.
The Vertical Axis in Higher Dimensions
If a third dimension is added, the x-y plane commonly lies flat, while the vertical axis is called the z-axis. A point (1,4,3) would therefore have a height of 3 above the x-y plane and projection onto the vertical axis of 3. The vertical axis may alternately represent a function of the two planar variable, f(x,y), with its value equaling the height of the point above the x-y plane.