The horizontal axis is reserved for the independent variable in a function. Time is always an independent variable in time-based functions.
However, duration can be dependent. It depends on what's being plotted.
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They are the axes. Usually horizontal = x-axis, vertical = y-axis. But that need not always apply. In a displacement-time or speed-time graph, for example, the horizontal axis = t-axis (for time).
Time is nearly always the independent variable, which are normally plotted on the horizontal axis.
The straight horizontal line on the graph says: "Whatever time you look at, the speed is always the same". This is the graph of an object moving with constant speed.
The independent variable, in this case time, is on the horizontal axis of a speed graph.
Distance and Time are variables and always moving. Therefore the answer is no. Let's suppose: If time is the vertical axis and distance (travelled) the horizontal axis. Standing still (not travelling) would show a vertical graph line. If distance is the vertical axis and time the horizontal axis. Then standing still would form a horizontal line based on time alone.