BJT in the forward-active region (Shockley): I_C = I_S exp(V_BE / V_T), V_T = 25.85 mV, I_S = 1e-14 A.
V_CE = V_CC - I_C R_C
saturation: V_CE <= 0.2 V -> I_C = (V_CC - 0.2)/R_C
cutoff: V_BE < ~0.45 V
gm = I_C / V_T ; A_v = -gm R_C (small signal)
load line: I_C = (V_CC - V_CE)/R_C
Small input signals are amplified with inverted phase; large ones clip against saturation and cutoff, and the exponential characteristic adds harmonic distortion (THD from a DFT of one period).