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

1.1

Change in Tandem

Rates of Change

Rates of Change in Linear and Quadratic Functions

Polynomial Functions and Rates of Change

Polynomial Functions and Complex Zeros

Polynomial Functions and End Behavior

Rational Functions and End Behavior

Rational Functions and Zeros

Rational Functions and Vertical Asymptotes

Rational Functions and Holes

Equivalent Representations of Polynomial and Rational Expressions

Transformations of Functions

Function Model Selection and Assumption Articulation

Function Model Construction and Application

Lesson

Diagnostic

after diagnostic

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 nonzero nth differences

over equal-spaced inputs a degree-n polynomial has constant nth differences, nonzero when the degree is exactly n (a lower degree makes them zero), so a table with them SUPPORTS a degree-n model. It does not pin down the function between or beyond the rows

Factored zero, multiplicity m

(x − a)^m: crosses if m odd, touches if m even

End behavior

leading term a_nxⁿ 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 → ±∞

Unit 1 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 14 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.