How much work is done by a spring when its end is moved from one position x_i to another x_f? Because the force is not constant, the work done is:

W = ∫_i^f F dx

Substituting -kx for F gives W = - ∫ kx dx,

This gives:   W = - ½ kx² | _xi ^x_f   = ½ kx_i² - ½ kx_f²