Problem 61142. Shifting vertically a function's graph

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Given a real function, f, by n input-output pairs, consider a translation in the up-down direction given by an amount k.
For a real constant k, you may assume that the translation is given. However, if k = 'mean', you firstly must determine the real constant k such that the translated function has an average value of 0 over the given interval (see figure below).
Find
y_shifted, which is the 1×n vector that stands for the outputs of the translated function;
v, which stands for either 'up' or 'down' if the function's graph is upward or downward shifted, respectively, and it stands for '' if the graph does not undergo a translation.
Hint. Calculate the mean of a piecewise linear discrete function, represented as an array of x and an array of y values. Be aware to the existence of calculus discrepancies whenever the function f will be continuous, but not piecewise linear.
input: (x, y, k)
output: [y_shifted, v]