An enzyme is a worker with a fixed number of hands. Give it a little substrate and it speeds up in step; flood it and it can’t go faster — every hand is full. That plateau is the shape of every enzyme in you. Slide the substrate (or tap) and watch the rate saturate.
Michaelis–Menten enzyme kinetics. The reaction rate v = Vmax·[S] / (Km + [S]) rises nearly linearly at low substrate and saturates toward Vmax as every active site fills — Km, the Michaelis constant, is exactly the substrate concentration at which the rate is half-maximal. Here Vmax=1 and Km=1, so at [S]=Km the rate is 0.5 and it approaches 1 as [S]→∞. The hyperbola and the current operating point are drawn live. A fail-loud self-check throws unless v(Km)=Vmax/2 and v→Vmax at saturation.
The classic single-substrate steady-state model (real enzymes show cooperativity, inhibition, multiple substrates). The hyperbola and the half-saturation identity are computed exactly.