  We construct an analytically solvable simplified model that captures the
essential features for primordial black hole (PBH) production in most models of
single-field inflation. The construction makes use of the Wands duality between
the constant-roll (or slow-roll) and the preceding ultra-slow-roll phases and
can be realized by a simple inflaton potential of two joined parabolas. Within
this framework, it is possible to formulate explicit inflationary scenarios
consistent with the CMB observations and copious production of PBHs of
arbitrary mass. We quantify the variability of the shape of the peak in the
curvature power spectrum in different inflationary scenarios and discuss its
implications for probing PBHs with scalar-induced gravitational wave
backgrounds. We find that the COBE/Firas $\mu$-distortion constraints exclude
the production of PBHs heavier than $10^4 M_\odot$ in single-field inflation.
