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 functionW(s), can another rationalM(s)(of simpler form) be constructed such that, for sufficiently large values of the constant multiplierBofM(s), the corresponding transient responses satisfy the condition,m(t) geq w(t)? 2)If so, can a range of values forBbe 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 ofM(s). Finally, a number of examples are given.