The 42nd Period | STEM Education Hub

Use this as a quick reference for polynomial vocabulary, local and global extrema, turning points between zeros, even-degree global extremes, and points of inflection.

Polynomial Functions and Rates of Change infographic

🧭 Plot Summary

Lesson 1.3 showed how the rates of a function can change. Now you put that to work on polynomial functions. You will name a polynomial's degree and leading coefficient, then find where it switches between increasing and decreasing. Those switches are local maximums and minimums, and the highest or lowest of them can be global. Between every two real zeros there must be one, every even-degree polynomial has a global extreme, and where the rates switch from increasing to decreasing (or the reverse) you find a point of inflection.

What you'll do in this lesson

Recognize polynomial functions and name the degree, leading term, and leading coefficient.
Find local maximums and minimums where a polynomial switches between increasing and decreasing.
Decide which local extrema are global, including endpoints of a restricted domain.
Show that there is a turning point between every two distinct real zeros.
Use even degree to know a global extreme exists, and find a quadratic's at its vertex.
Locate points of inflection where the rates switch from increasing to decreasing, or the reverse.

Why it matters

Maximizing profit, minimizing cost, and finding the peak of a launched object all come down to locating extrema. Reading a graph's turning points and bends is also how you will sketch and analyze polynomials in Lessons 1.5 and 1.6, and it is the idea behind the derivative later on.

✅ Self-Check Before You Move On

Check off each item as you get there. These aren't grades — they're your own signal.

I can recognize a polynomial function and name its degree, leading term, and leading coefficient.I can find local maximums and minimums where a polynomial switches between increasing and decreasing, including at the included endpoints of a restricted domain.I can tell a global maximum or minimum from a local one.I can explain why there is a local maximum or minimum between every two distinct real zeros.I can tell that an even-degree polynomial has a global maximum or a global minimum, and find a quadratic's at its vertex.I can find points of inflection from where the rates of change switch between increasing and decreasing.

🔬 Deep Dive

Your textbook for this topic: polynomial vocabulary, local and global extrema, turning points between zeros, even-degree extremes, and points of inflection, with interactive tools.

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🧭 Guided Practice

Step through worked examples for every essential knowledge statement in 1.4.A. At least three examples each, revealed one step at a time.

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📝 Assignment

20 questions covering the whole lesson. Pick A, B, C, or D and find out right away: a correct answer shows the full worked solution, and a wrong one tells you the most likely slip.

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📌 AP Exam Note

Keep three ideas apart. A local extremum is where the outputs switch between increasing and decreasing, a global extremum beats every other output, and a point of inflection is where the rates switch between increasing and decreasing, not the outputs. Support each claim with rates of change or a sign change, and remember that an endpoint of a restricted domain can be a local or global extremum.