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AP Pre-Calculus Unit 1 Concavity, end behavior & function notes

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These notes cover the foundations of polynomial behavior in precalculus. They begin with several factoring examples, including difference of squares, factoring by grouping, and breaking down higher-degree expressions. After the warm-up factoring problems, the notes explain how zeros of a polynomial function work: plugging a root into the function results in zero, and an n-degree polynomial has n total zeros when counted with multiplicity. A key idea highlighted here is multiplicity—roots with even multiplicity cause the graph to bounce off the x-axis, while roots with odd multiplicity cause the graph to cross it. The sketches on page 2 illustrate this idea and also show that complex zeros always come in conjugate pairs. The notes then move into end behavior. Using sketches and limit notation, they show how the degree and the sign of the leading coefficient determine whether the graph rises or falls on each side. Even-degree polynomials either open up or down on both ends, while odd-degree polynomials rise on one side and fall on the other. Several examples on page 3 demonstrate this with limits approaching infty or -infty. Later sections introduce finite differences as a way to identify the degree of a polynomial from a table of values. Equal spacing in x-values is required, and constant second differences indicate a quadratic pattern. The final pages include small sketches analyzing increasing and decreasing intervals, concavity, and rate of change. The notes point out that concave-up curves produce average-rate estimates that are less than the true value, while concave-down curves produce estimates greater than the true value. The last examples also contrast even and odd functions, showing that even functions produce the same y-values at opposite x-values, while odd functions produce opposite y-values.

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Institution
Senior / 12th Grade
Course
AP Pre-Calculus









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Institution
Senior / 12th grade
Course
AP Pre-Calculus
School year
4

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Uploaded on
November 30, 2025
Number of pages
5
Written in
2025/2026
Type
Class notes
Professor(s)
Novick
Contains
All classes

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