Options Strategy Question: understanding max profits/loss and breakeven

Hi All, Can anyone help explain how to derive the maximum profit/loss and breakeven prices formulas? That is, what’s the best way to go about retaining these different formulas for the below strategies without blindly memorizing them? For the deriving the profit, once I know what the strategy involves (for example, bull call spread is long a call at low exercise price and short a call at high exercise price), then the profit can be derived as the sum of the long call max(0,S_t - X_low) and short call max(0,X_high-S_t). But I can’t seem to make the connection to deriving the maximum profit/loss and breakeven prices. Any thoughts? -covered call -protective put - bull call spread - bear call spread - bear put spread - butterfly spread with calls - butterfly spread with puts

bull spread profit: max(0,S-X1) - Max(0,S-X2) - c1 + c2 splitting up into ranges: Payoff will be: S < X1 --> 0 X1 < S < X2 --> S-X1 - 0 = S-X1 S > X2 -> S - X1 - (S-X2) = X2 - X1 Max Profit - will be the X2 - X1 - c1 + c2 (look at the place where there is no S) Break even: for breakeven --> look at the only payoff situation which has S anywhere in it. it is where X1 < S < X2 --> Payoff is S-X1 Profit then is S-X1 - c1 + c2 if this profit is 0 – for a breakeven S=X1+C1 - C2

If you have in your mind, how the particular strategy will look, then it is easier to solve the problem. For eg: Think about how a covered call will look on a graph. If you can visualize that, then it gets easy.