Energy transformation diagrams, like the one in the picture on the left, are used to show how energy is transferred throughout a series of movements. Energy can not be created or destroyed but can be changed into unusable forms of energy. That being said energy is not destroyed throughout the flight path of the shuttlecock but rather lost to unusable forms of energy such as sound or elastic energy.
Gravitational potential energy is the amount of energy that an object can possibly get from gravity. Gravitational potential energy depends on the a=mass of the object, acceleration due to gravity which is constant on earth(9.8m/s squared) and the height at which the object is at. The calculation on the right show the amount of gravitational potential energy that the birdie has when it is at its maximum height. Now this is not to say that when the birdie falls it will have that amount of gravitational energy but rather that there is potential for there to be that much gravitational energy.