The x axis is horizontal and the y axis is vertical.
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For an example, if negative and positive directions on the X and Y axes represent left/right and forward/backward, Z axis represents up/down.
A point on a 3-d coordinate system would take the form of (x,y,z). You go x units on the x-axis (left or right), y units on the y-axis (up or down), and z units on the z-axis, (front or back).
In 2-dimensions, it is the y-axis. In 3-dimensions, though, the x and y axes form the horizontal plane and the z-axis is the vertical axis.
The side is the WRONG answer. A Y axis is up and down and a X axis is left and right. Easiest way to remember is Y to the sky, X to the side. Also, the Z axis is front to back.
No. An axis normally stretches out in both directions - infinitely. The x and y axis define a plane and the z-axis is normally at right angles to that plane. A good way to visualise it is to look at a corner of a room. If you face the corner, the convention is that the floor and the wall to the left meet along the x-axis, the floor and the right wall at the y-axis, and the two walls along the z-axis.