Polynomial and Rational Functions
Fourteen topics on how functions change and how their structure shows on a graph. Rates of change and concavity, polynomial zeros with their multiplicities and complex pairs, end behavior from the leading term, rational functions with their asymptotes and holes, equivalent forms, transformations, and the discipline of choosing and building function models.
Topics
Key forms For every problem in this unit
Average rate of change on [a, b]
(f(b) − f(a)) / (b − a)
Concave up
rate of change is increasing
Concave down
rate of change is decreasing
Constant nth differences
degree-n polynomial (equal-spaced inputs)
Factored zero, multiplicity m
(x − a)m: crosses if m odd, touches if m even
End behavior
leading term anxn decides both ends
Vertical asymptote
zero of denominator that does NOT cancel
Hole
common factor of numerator and denominator
Horizontal asymptote
compare degrees; ratio of leading terms as x → ±∞