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A method involving root-locus techniques is developed by which one can analyze transient responses by ascertaining bounds on them. In particular, the following questions are considered: 1) Given a rational system function , can another rational (of simpler form) be constructed such that, for sufficiently large values of the constant multiplier of , the corresponding transient responses satisfy the condition, If so, can a range of values for be determined for which the same condition holds? Necessary conditions for an affirmative answer to question 1) are first developed and then sufficient conditions are obtained. Then, a general method using root-locus techniques is developed for answering question 2). Certain special cases are studied in detail and necessary and sufficient conditions are obtained, thus leading to the best possible bound for the given form of . Finally, a number of examples are given.