Time value of an American call across stock prices

Take an American call written on a stock that pays no dividends. Graph its time value, i.e. option price minus intrinsic value $\max(S-K,0)$, as a function of the current stock price, and explain the shape.

A tent peaking at the strike. Time value is near zero for deep out-of-the-money calls (little chance of payoff), peaks at $S = K$ where uncertainty matters most, and for deep in-the-money calls falls toward $K(1 - e^{-rT})$, not zero, because the holder still earns interest by deferring payment of the strike. Since it stays positive, early exercise is never optimal.