Why do we choose - 2, - 1, 0, 1, 2 for the x values?

In many cases, the function is easily evaluated and/or graphed for those x-values.

Step-by-step explanation:

We are usually interested in x-values that relate to the problem at hand. In many cases, the functions we study are graphed on a coordinate plane that includes the origin and a few numbers in every direction. That is, x-values of - 2, - 1, 0, 1, 2 are right in the middle of the graph we want to create.

Another reason for choosing small values of x is that we may want to raise these values to some power (as for evaluating a polynomial). Many of us have memorized the first few powers of the first few integers, so using small integers makes the evaluation easier.

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That is not always the case. Some functions may not be defined for negative values of x, or those values may not exist in the "practical domain" of the function. Trigonometric functions may be more easily evaluated for multiples of π/6, instead of small integers. Other functions may be scaled or offset so that small integer values of x are of no interest.