-- constant acceleration
-- speed increasing at a steady rate
-- distance increasing as the square of the time since everything started
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Constant speed ... zero acceleration.
The graph you described is a speed-time plot. If the line is horizontal, that indicates no change in speed over time. In other words, there is no acceleration (acceleration is zero), since there is no change in speed.
constant speed
that would indicate that the object is at rest (static object) :D
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.