Polynomials and ROC — AP Precalculus | The Complete Equation

AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS

1.4 — Polynomial Functions and Rates of Change

Notes — Extrema, Inflection Points, and Behavior

💡 Learning Objectives (1.4.A)

By the end of this lesson you will be able to:

Identify and define a polynomial function in standard form, naming its degree, leading term, and leading coefficient
Locate local maxima and minima from a graph or formula
Distinguish local extrema from global (absolute) extrema
Recognize that a local extremum sits between any two distinct real zeros
Identify points of inflection where concavity changes
Conclude that an even-degree polynomial has either a global maximum or a global minimum

1. Polynomial Functions

💡 Definition

A nonconstant polynomial function of x has the form

p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₂x² + a₁x + a₀,

where n is a positive integer, every aᵢ is real, and the leading coefficient aₙ is nonzero. The integer n is the degree of the polynomial. A constant function (e.g., p(x) = 7) is also called a polynomial — of degree 0.

Degree	Common name	Example
0	Constant	p(x) = 4
1	Linear	p(x) = 3x − 2
2	Quadratic	p(x) = x² − 4x + 1
3	Cubic	p(x) = x³ + 2x − 5
4	Quartic	p(x) = x⁴ − 3x² + 1
5	Quintic	p(x) = 2x⁵ − x

2. Local Maxima and Minima

As you walk along a polynomial graph from left to right, the curve goes up, comes down, goes up again — and so on. The peaks and valleys are called extrema.

💡 Defining extrema

Local maximum: a point where f's value is greater than the values at every nearby input on both sides.
Local minimum: a point where f's value is less than the values at every nearby input on both sides.
Global (absolute) maximum: the largest value of f on its entire domain.
Global (absolute) minimum: the smallest value of f on its entire domain.

A quartic with two local minima and one local maximum. The lower of the two minima is the global minimum. There is no global maximum here — the curve climbs without bound on both ends.

⚠️ Local vs. global

Every global extremum is automatically a local extremum at the same point, but the converse fails. A local maximum can be much smaller than a global maximum elsewhere on the graph.

3. A Useful Theorem: Extrema Between Zeros

💡 Between every two distinct real zeros, an extremum lurks

If a polynomial function p has two distinct real zeros at x = a and x = b with a < b, then somewhere strictly between a and b, p achieves a local maximum or local minimum.

Why? Because p is zero at both a and b but nonzero between them, the graph must rise above (or dip below) the x-axis in between, and somewhere in that excursion the graph reaches a peak (or trough).

A polynomial with three distinct real zeros and two extrema — one local maximum between the first two zeros, one local minimum between the last two.

4. Even Degree ⇒ a Global Extremum

💡 A guaranteed global extremum

A polynomial of even degree must have either a global maximum or a global minimum (depending on the sign of the leading coefficient).

Even degree, positive leading coefficient: both ends of the graph go to +∞, so there is a global minimum somewhere.
Even degree, negative leading coefficient: both ends go to −∞, so there is a global maximum.

Polynomials of odd degree have no global maximum and no global minimum — both ends of the graph head off to infinity in opposite directions.

5. Inflection Points and Rate-of-Change Behavior

Recall from Topic 1.1 that an inflection point is a point where the concavity of the graph switches direction. For a polynomial:

If the rate of change is increasing on an interval, the graph is concave up there.
If the rate of change is decreasing on an interval, the graph is concave down there.
An inflection point is a point on the graph where the rate of change shifts from increasing to decreasing or vice versa.

A cubic has exactly one inflection point. To the left of x = 1 the graph is concave down; to the right it is concave up.

📘 Example — Reading inflection from a graph

Consider the curve y = (x − 1)³ shown above:

• On (−∞, 1): graph is concave down (rate of change is decreasing — the slopes go from very large negative to small).

• On (1, ∞): graph is concave up (rate of change is increasing again).

• At x = 1: concavity changes — this is the inflection point.

6. Putting It Together — Reading a Polynomial Graph

Given the graph of a polynomial, you should be able to identify:

The (real) zeros — x-intercepts where the curve meets the x-axis
All local maxima and minima
Whether each extremum is global or only local
Intervals of increase / decrease
Intervals of concave up / concave down
Any inflection points
The end behavior (which way each end goes)

📘 Example — A worked sketch

Suppose you are told that p is a polynomial with these features:

• zeros at x = −2, x = 1, x = 3;

• ends both head to +∞ (so even degree, positive leading coefficient);

• a local maximum near x = 0 and a local minimum near x = 2.

Then a quartic such as p(x) = (x + 2)(x − 1)²(x − 3) (with degree 4 once expanded) is a possible model. The two local minima theorem requires you to count multiplicities.

7. Summary

A polynomial of degree n has standard form aₙxⁿ + … + a₀ with aₙ ≠ 0
Local extrema are peaks/valleys; global extrema are the largest/smallest output values overall
Between any two distinct real zeros, there must be at least one local extremum
Even-degree polynomials always have a global maximum or a global minimum
Inflection points are where concavity changes; equivalently, where the rate of change switches from increasing to decreasing or vice versa