Precalculus · Unit 1: Polynomial and Rational Functions · 13 min read · Updated 2026-07-21
Polynomial functions and rates of change — AP Precalculus
AP Precalculus · Unit 1: Polynomial and Rational Functions · 13 min read
1. What a Polynomial Function Is★☆☆☆☆⏱ 3 min
Everything in this topic is about **polynomial** functions, so it is worth pinning down exactly what counts as one. A nonconstant polynomial function of $x$ is any function that can be written in the general form below — a sum of terms, each a real number times a whole-number power of $x$.
Here $n$ is a **positive integer**, each coefficient $a_i$ is a real number, and the leading coefficient $a_n$ is **not zero**. Three names come straight out of this form:
One edge case is worth remembering: a **nonzero constant** function such as $p(x) = 6$ is also a polynomial function, of **degree zero**.
2. Local (Relative) Maximum and Minimum Values★★☆☆☆⏱ 4 min
From Topic 1.2 the **average rate of change** over an interval is the change in output divided by the change in input, and from Topic 1.3 its **sign** tells you the direction: a positive rate means the function is increasing there, a negative rate means it is decreasing. A polynomial's key features sit exactly where that direction changes.
There is a second place a local extreme value can appear. If the polynomial's **domain is restricted** and the endpoint is **included**, that endpoint is a local maximum or minimum as well — the function simply has nowhere further to go on that side.
3. Global (Absolute) Maximum and Minimum Values★★☆☆☆⏱ 3 min
A local extreme value becomes a **global** one when it wins against the whole function, not just its neighbours.
Degree tells you when a global extreme value is guaranteed. A polynomial function of **even degree** always has either a global maximum or a global minimum. For a **quadratic** function, that global maximum or minimum occurs at the **vertex**.
4. What Two Distinct Real Zeros Force★★☆☆☆⏱ 2 min
The real zeros of a polynomial are the input values where its output is $0$ — the inputs where the graph meets the $x$-axis. Knowing two of them tells you something about the shape in between, with no computation at all.
The reason is easy to picture: leaving one zero the graph moves off the $x$-axis, and to arrive at the other zero it has to come back. It cannot do both while only rising, or only falling — somewhere in between it must turn around.
5. Points of Inflection★★★☆☆⏱ 3 min
So far we have used the **sign** of the rate of change. The last characteristic comes from asking a different question about the same numbers: is the rate of change itself getting bigger or smaller?
That is the concavity idea from Topic 1.3. Where the rate of change is **increasing**, the graph is **concave up**; where the rate of change is **decreasing**, the graph is **concave down**. The place where that switches over has a name.
Common Pitfalls
Why: A global maximum must be greater than **all other output values** of the function. Many polynomials, such as $f(x)=x^3-3x$, have local extreme values but no global ones at all because the outputs are unbounded.
Why: At an **included** endpoint of a restricted domain the polynomial has a local maximum or minimum, even though the function never switches direction there.
Why: In $p(x) = 5 - 2x^3 + x$ the first term is $5$, but the leading term is the one with the greatest exponent, $-2x^3$.
Why: That describes a local maximum or minimum. A point of inflection is where the **rate of change** switches from increasing to decreasing or the reverse — the function itself can be increasing on both sides of it.
Why: Two distinct real zeros already force at least one local maximum or minimum strictly between them, because the graph must leave the axis and return to it.