Chapter 3 Polynomial and Rational Functions

Polynomial and rational functions describe curved relationships that linear models cannot capture. Quadratic models represent areas and trajectories, higher-degree polynomials can model responses with several turning points, and rational functions describe ratios whose behaviour may change sharply near excluded inputs. This chapter develops the graphical features emphasized in the MATH 128 notes: vertex form, completing the square, intercepts, local extrema, end behaviour, zeros and multiplicity, asymptotes, holes, and systematic graphing. Water-engineering examples include basin dimensions, pump-response curves, and models whose denominators represent operational limits. Algebraic work is connected continually to graph shape and physical interpretation.

Learning outcomes. By the end of this chapter, you should be able to:

• graph quadratic functions from vertex, standard, and factored forms;
• complete the square and interpret the resulting vertex;
• describe polynomial end behaviour and the effect of zero multiplicity;
• produce a systematic sketch of a polynomial function;
• find domains, intercepts, holes, and asymptotes of rational functions; and
• sketch and interpret rational functions in applied settings.