Cold water's temperature increase while hot water's temperature decrease.
The rate of increase/decrease is different. The rate of decrease is faster than the rate of increase as metal is a better thermal energy conductor, causing thermal energy to be lost to the cold water at a faster rate. The upper curve showed faster rate of flow of TE.
Why is metal and styrofoam cup used respectively?
MetalCup---Allow speedier transfer of thermal energy from the hot water to the cold water so as to speed up the rate of thermal energy gain and loss
StyrofoamCup---Reduce(not prevent)the cold water from gaining heat from surroundings. This ensure that room temperature does not play as a factor to affect the experiment.
When the question is little open, you need to make assumptions so that you can narrow the possibilities.
The hot water is losing TE from the room temperature and the cold water. Visa Versa; the cold water is gaining TE from the room temperature and the hot water. In other words, using styrofoam cup help eliminate the room temperature.