Polynomial functions and rates of change

AP Precalculus· Unit 1 · Polynomial & 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 $x$ .

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:

📘 Definition

Degree, leading term, leading coefficient

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

The degree is $n$ , the greatest exponent appearing. The leading term is $a_{n} x^{n}$ , the term carrying that greatest exponent. The leading coefficient is $a_{n}$ , the number multiplying it.

📐 Worked Example

For $p (x) = 7 + x^{2} - 4 x^{5}$ , 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 $- 4 x^{5}$ , so that is the leading term, and its coefficient $- 4$ is the leading coefficient.
5
Answer: degree $5$ , leading term $- 4 x^{5}$ , 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.

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.

📘 Definition

Local (relative) maximum or minimum

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

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.

📐 Worked Example

For $f (x) = x^{3} - 3 x$ , 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 $Δ x = 1$ ):
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 $f (x) = x^{3} - 3 x$ is now given only on the restricted domain $- 2 \leq x \leq 0$ . 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 $(0, 0)$ .

✓ Quick check

Test your understanding with this multiple-choice question:

A polynomial $g$ has average rates of change $- 6, - 1, + 3, + 5$ over four consecutive equal-length intervals covering $2 \leq x \leq 6$ . 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
Reveal answer
1 —
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.

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.

📘 Definition

Global (absolute) maximum and minimum

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

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.

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 $x^{2} \geq 0$ and therefore $- x^{2} \leq 0$ 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.

✓ Quick check

Test your understanding with this multiple-choice question:

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
Reveal answer
2 —
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.

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.

📘 Definition

A turn between two zeros

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

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.

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.

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.

📘 Definition

Point of inflection

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

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.

📐 Worked Example

Using the same rates of change for $f (x) = x^{3} - 3 x$ over $- 2 \leq x \leq 2$ , 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 $(0, 0)$ .

✓ Quick check

Test your understanding with this multiple-choice question:

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
Reveal answer
1 —
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.

6. Common Pitfalls

Wrong move:

Calling the highest turn on the visible graph the global maximum.

Why:

A global maximum must be greater than all other output values of the function. Many polynomials, such as $f (x) = x^{3} - 3 x$ , have local extreme values but no global ones at all because the outputs are unbounded.

Correct move:

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 move:

Ignoring the endpoints when the domain is restricted.

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.

Correct move:

List the interior switches and every included endpoint as candidates for local extreme values.

Wrong move:

Reading the leading coefficient from whichever term is written first.

Why:

In $p (x) = 5 - 2 x^{3} + x$ the first term is $5$ , but the leading term is the one with the greatest exponent, $- 2 x^{3}$ .

Correct move:

Reorder by descending power first, then take the degree, leading term, and leading coefficient from the highest-degree term.

Wrong move:

Treating a point of inflection as a place where the function changes direction.

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.

Correct move:

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 move:

Concluding nothing about a polynomial's shape when you only know its zeros.

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.

Correct move:

Pair up consecutive distinct real zeros and record that at least one local extreme value lies between each pair.

7. Quick Reference Cheatsheet

Concept	Definition / Rule	Key Notes
Polynomial function	$p (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ , $n$ 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	$a_{n} x^{n}$ / $a_{n}$	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.