There are two very separate issues here, and I think it makes sense to disentangle them first. Unless I am told otherwise, if I am given some data of rate versus substrate concentration, I will assume:
(1) We are working under initial velocity conditions, meaning that perhaps 1-3% of the substrate has been consumed (converted into product). Therefore depletion of substrate has not occurred to any significant extent and the rate will not go down.
(2) We are working under steady-state conditions. For many situations, steady state is reached in less than a second. Steady-state just means that the concentration of the complex between the enzyme and substrate, [E•S], is not changing much in time.
If both of these conditions are met, then the graph of rate versus concentration of substrate will be a rectangular hyperbola, meaning linear at low concentration of substrate and gradually leveling off. When substrate concentration is very high, all of the enzyme active sites are occupied (the enzyme is "saturated"), and higher concentrations of substrate can no longer affect the rate.