# Polynomial functions and rates of change

> AP Precalculus · Unit 1: Polynomial and Rational Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u1-polynomial-functions-and-rates-of/

What a polynomial function is — its degree, leading term, and leading coefficient — and the key characteristics you can read from its rate of change: where it has local (relative) maximum and minimum values, including at an included endpoint of a restricted domain; when such a value is a global (absolute) maximum or minimum; what two distinct real zeros force to happen between them; what an even degree guarantees; and where its points of inflection are.

**Prerequisites:** Average rate of change over an interval (Topic 1.2); Reading increasing and decreasing behaviour from the sign of the rate of change (Topic 1.3); Concave up and concave down described by whether the rate of change is increasing or decreasing (Topic 1.3)

## Learning objectives

- Write a polynomial function in general form and identify its degree, leading term, and leading coefficient.
- Locate local (relative) maximum and minimum values where a polynomial switches between increasing and decreasing, including at an included endpoint of a restricted domain.
- Decide when a local extreme value is a global (absolute) maximum or minimum, and recognise that an even-degree polynomial always has one.
- Use two distinct real zeros to conclude that a local maximum or minimum lies between them.
- Locate points of inflection where the rate of change switches from increasing to decreasing or from decreasing to increasing.

## What a Polynomial Function Is

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 $x$.

$$p(x) = a_n x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_2x^2 + a_1x + a_0$$

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:

**Degree, leading term, leading coefficient** — The **degree** is $n$, the greatest exponent appearing. The **leading term** is $a_nx^n$, the term carrying that greatest exponent. The **leading coefficient** is $a_n$, the number multiplying it.

*Notation:* \deg(p)=n,\quad \text{leading term } a_nx^n,\quad \text{leading coefficient } a_n

> **tip**
>
> The leading term is the one with the **highest power**, not the one written first. In $p(x) = 5 - 2x^3 + x$ the leading term is $-2x^3$ and the leading coefficient is $-2$ — reorder the terms by descending power before reading them off.

**Worked example:** For $p(x) = 7 + x^2 - 4x^5$, state the degree, the leading term, and the leading coefficient.

1. Write the terms in descending order of exponent so the highest power comes first:
2. $$p(x) = -4x^5 + x^2 + 7$$
3. The greatest exponent is $5$, so the degree is $5$.
4. The term carrying that exponent is $-4x^5$, so that is the leading term, and its coefficient $-4$ is the leading coefficient.
5. **Answer:** degree $5$, leading term $-4x^5$, leading coefficient $-4$.

One edge case is worth remembering: a **nonzero constant** function such as $p(x) = 6$ is also a polynomial function, of **degree zero**.

## Local (Relative) Maximum and Minimum Values

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.

**Local (relative) maximum or minimum** — Where a polynomial function **switches between increasing and decreasing**, it has a local — or relative — maximum or minimum output value. Increasing then decreasing gives a local **maximum**; decreasing then increasing gives a local **minimum**.

*Notation:* +\ \rightarrow\ -\ :\ \text{local max}\qquad -\ \rightarrow\ +\ :\ \text{local min}

**Worked example:** For $f(x) = x^3 - 3x$, use average rates of change over the equal-length intervals from $x=-2$ to $x=2$ to locate the local maximum and local minimum.

1. Evaluate $f$ at equally spaced inputs:
2. $$\begin{array}{c|ccccc} x & -2 & -1 & 0 & 1 & 2\\ \hline f(x) & -2 & 2 & 0 & -2 & 2 \end{array}$$
3. Find the average rate of change across each step (each has $\Delta x = 1$):
4. $$\frac{2-(-2)}{1}=+4,\qquad \frac{0-2}{1}=-2,\qquad \frac{-2-0}{1}=-2,\qquad \frac{2-(-2)}{1}=+4$$
5. The rates run $+4,\ -2,\ -2,\ +4$. The sign turns from positive to negative near $x=-1$: the function stops increasing and starts decreasing, so there is a **local maximum** at $(-1, 2)$.
6. The sign turns from negative to positive near $x=1$, so there is a **local minimum** at $(1, -2)$.

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.

**Worked example:** The same $f(x)=x^3-3x$ is now given only on the restricted domain $-2 \le x \le 0$. Identify every local maximum and local minimum.

1. On this domain $f$ increases from $x=-2$ to $x=-1$ and decreases from $x=-1$ to $x=0$.
2. The switch from increasing to decreasing at $x=-1$ still gives a **local maximum** at $(-1, 2)$.
3. Both endpoints are included, so each is also a local extreme value: $x=-2$ gives a **local minimum** at $(-2,-2)$, and $x=0$ gives a **local minimum** at $(0,0)$.
4. **Answer:** local maximum at $(-1,2)$; local minima at the included endpoints $(-2,-2)$ and $(0,0)$.

> **warning**
>
> The word *local* is doing real work: it only compares a point with the inputs **near** it. A local maximum says nothing on its own about the function's largest output overall — that is the next section.

**Check your understanding**

Test your understanding with this multiple-choice question:

1. A polynomial $g$ has average rates of change $-6,\ -1,\ +3,\ +5$ over four consecutive equal-length intervals covering $2 \le x \le 6$. Which is true?

   - $g$ has a local maximum in this window
   - $g$ has a local minimum in this window
   - $g$ has both a local maximum and a local minimum here
   - $g$ has neither

   *Answer:* $g$ has a local minimum in this window

   *Why:* The rates change sign once, from negative to positive: the function stops decreasing and starts increasing, which is a local minimum. There is no positive-to-negative switch, so there is no local maximum in this window.

## Global (Absolute) Maximum and Minimum Values

A local extreme value becomes a **global** one when it wins against the whole function, not just its neighbours.

**Global (absolute) maximum and minimum** — If a local maximum is **greater than all other output values** of the function, that local maximum is a **global**, or **absolute**, maximum. Likewise, if a local minimum is **less than all other output values**, it is a global, or absolute, minimum.

*Notation:* \text{global max } f(c)\ :\ f(c) > f(x)\ \text{ for every other } x \text{ in the domain}

> **warning**
>
> **Not every polynomial has one.** For $f(x)=x^3-3x$ the outputs climb without bound in one direction and fall without bound in the other, so it has a local maximum and a local minimum but **no global maximum and no global minimum**. Never assume the highest turn on the part of the graph you can see is the global maximum.

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**.

**Worked example:** Explain why $g(x) = -x^2$ has a global maximum, and state where it occurs.

1. Squaring any real number gives a value that is zero or positive, so $x^2 \ge 0$ and therefore $-x^2 \le 0$ for every input.
2. The value $0$ is actually attained, at $x=0$: $g(0) = 0$.
3. So the output $0$ is greater than every other output value of $g$ — that local maximum is the **global (absolute) maximum**.
4. $g$ is a quadratic, and as expected the global maximum sits at its **vertex**, the point $(0,0)$. Its degree $2$ is even, matching the guarantee that an even-degree polynomial has a global maximum or a global minimum.

**Check your understanding**

Test your understanding with this multiple-choice question:

1. Which statement is guaranteed to be true for **every** polynomial function of degree $4$?

   - It has a global maximum
   - It has a global minimum
   - It has either a global maximum or a global minimum
   - It has both a global maximum and a global minimum

   *Answer:* It has either a global maximum or a global minimum

   *Why:* Degree $4$ is even, and an even-degree polynomial function has either a global maximum or a global minimum — which one depends on the function. It cannot have both: the outputs are unbounded in one direction.

## What Two Distinct Real Zeros Force

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.

**A turn between two zeros** — Between every two **distinct real zeros** of a nonconstant polynomial function there must be **at least one** input value corresponding to a local maximum or a local minimum.

*Notation:* p(a)=p(b)=0,\ a \ne b \;\Rightarrow\; \text{a local max or min lies strictly between } a \text{ and } b

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.

**Worked example:** A polynomial $p$ has real zeros at $x=-4$, $x=1$, and $x=6$, and no others. What can you conclude about its local extreme values?

1. Take the zeros in consecutive pairs. Between $x=-4$ and $x=1$ there must be at least one local maximum or local minimum.
2. Between $x=1$ and $x=6$ there must again be at least one local maximum or local minimum.
3. **Answer:** $p$ has at least two local extreme values, one in $-4 < x < 1$ and one in $1 < x < 6$. Their exact locations and whether each is a maximum or a minimum cannot be settled from the zeros alone.

> **tip**
>
> The zeros must be **distinct**. A repeated zero does not give you two separate crossings, so it forces nothing about a turn between them.

## Points of Inflection

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.

**Point of inflection** — A **point of inflection** of a polynomial function occurs at an input value where the **rate of change of the function changes from increasing to decreasing, or from decreasing to increasing**. Equivalently, it is where the graph changes from **concave up to concave down**, or from **concave down to concave up**.

*Notation:* \text{rate of change: increasing} \leftrightarrow \text{decreasing} \;\Longleftrightarrow\; \text{concave up} \leftrightarrow \text{concave down}

**Worked example:** Using the same rates of change for $f(x)=x^3-3x$ over $-2 \le x \le 2$, locate the point of inflection.

1. The average rates of change over the four equal steps were $+4,\ -2,\ -2,\ +4$.
2. Reading them left to right, the rate first **falls** ($+4$ down to $-2$): over that stretch the rate of change is decreasing, so the graph is **concave down**.
3. Then the rate **rises** ($-2$ back up to $+4$): the rate of change is increasing there, so the graph is **concave up**.
4. The rate of change switches from decreasing to increasing at $x = 0$, so there is a **point of inflection** at $(0, 0)$.

> **warning**
>
> Do not confuse the two switches. Where the **function** switches between increasing and decreasing you get a local maximum or minimum. Where the **rate of change** switches between increasing and decreasing you get a point of inflection. At a point of inflection the function itself may still be increasing the whole time.

**Check your understanding**

Test your understanding with this multiple-choice question:

1. A polynomial $h$ has average rates of change $+1,\ +4,\ +9,\ +7,\ +2$ over five consecutive equal-length intervals. Which is true?

   - $h$ has a local maximum somewhere in this window
   - $h$ has a point of inflection somewhere in this window
   - $h$ is concave up across the whole window
   - $h$ is decreasing somewhere in this window

   *Answer:* $h$ has a point of inflection somewhere in this window

   *Why:* Every rate is positive, so $h$ is increasing throughout — no local extreme value and never decreasing. But the rates rise ($1,4,9$) and then fall ($9,7,2$): the rate of change switches from increasing to decreasing, which is a point of inflection, with the graph turning from concave up to concave down.

## Common pitfalls

- **Wrong:** Calling the highest turn on the visible graph the global maximum.
  - Why it fails: 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.
  - Correct: Check the whole function, not the visible window. Only claim a global maximum or minimum when nothing else the function outputs can beat it — an even degree is what guarantees one exists.
- **Wrong:** Ignoring the endpoints when the domain is restricted.
  - Why it fails: At an **included** endpoint of a restricted domain the polynomial has a local maximum or minimum, even though the function never switches direction there.
  - Correct: List the interior switches **and** every included endpoint as candidates for local extreme values.
- **Wrong:** Reading the leading coefficient from whichever term is written first.
  - Why it fails: In $p(x) = 5 - 2x^3 + x$ the first term is $5$, but the leading term is the one with the greatest exponent, $-2x^3$.
  - Correct: Reorder by descending power first, then take the degree, leading term, and leading coefficient from the highest-degree term.
- **Wrong:** Treating a point of inflection as a place where the function changes direction.
  - Why it fails: 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.
  - Correct: Ask which quantity is switching: the function's direction gives a local extreme value; the rate of change's direction gives a point of inflection.
- **Wrong:** Concluding nothing about a polynomial's shape when you only know its zeros.
  - Why it fails: 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.
  - Correct: Pair up consecutive distinct real zeros and record that at least one local extreme value lies between each pair.

## Cheatsheet

| Concept | Definition / Rule | Key Notes |
| --- | --- | --- |
| Polynomial function | $p(x)=a_nx^n+\cdots+a_1x+a_0$, $n$ a positive integer, $a_n \neq 0$ | A nonzero constant is a polynomial of degree $0$ |
| Degree | The greatest exponent $n$ appearing | Reorder by descending power before reading it |
| Leading term / leading coefficient | $a_nx^n$ / $a_n$ | From the highest-degree term, not the first written |
| Increasing / decreasing | Rate of change positive / negative | The sign gives the direction |
| Local (relative) max or min | Where the function switches increasing $\leftrightarrow$ decreasing | Also at an **included endpoint** of a restricted domain |
| Global (absolute) maximum | A local max greater than all other output values | Not every polynomial has one |
| Global (absolute) minimum | A local min less than all other output values | Not every polynomial has one |
| Even degree | Has either a global maximum or a global minimum | For a quadratic it occurs at the vertex |
| Two distinct real zeros | At least one local max or min lies strictly between them | The graph must leave the axis and return |
| Point of inflection | Where the rate of change switches increasing $\leftrightarrow$ decreasing | Same as concave up $\leftrightarrow$ concave down |

## What's next

You can now read a polynomial's key characteristics from its rate of change: where it turns, which turns are global, what its zeros force in between, and where its concavity flips. Topic 1.5, Polynomial Functions and Complex Zeros, looks at those zeros directly — how many a polynomial of degree $n$ has, what multiplicity means for the graph, and how non-real zeros come in conjugate pairs.

- [](https://www.owlsprep.com/study/ap-precalculus-u1-polynomial-functions-and-complex-zeros/)

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