AP Precalc 1.4 Polynomials & ExtremaAP Precalc 1.4 Polynomials & Extrema

A P P R E C A L C U L U S · UNIT1

Polynomial Functions and Rates of Change

Topic 1.4 Study Guide

In one sentence: a polynomial's degree and leading coefficient tell you how many turns it can have and whether it has an absolute maximum or minimum.

Polynomial vocabulary

p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀

Degree = largest exponent. Leading term = the term with that exponent. Leading coefficient = its number.

Not in standard form? Hunt for the biggest exponent; do not trust the first term. Factored form: multiply the leading terms of the factors. Degree = sum of exponents, leading coefficient = product of the coefficients.

Example. f(x) = 14x² − 6x⁵ + 2x − 1: degree 5, leading coefficient −6.

Factored. g(x) = 4(x − 1)²(3 − x): leading term 4·x²·(−x) = −4x³. Degree 3, leading coefficient −4. Extrema

Term Meaning

Local (relative) max Switches from increasing to decreasing

Local (relative) min Switches from decreasing to increasing

Absolute (global) max / min Greatest local max / least local min over the whole domain An included endpoint of a restricted domain can also be a local extremum. Always give both parts: the output value and where it occurs (the x-value).

Key rules

Two zeros ⇒ an extremum. Between two zeros of a nonconstant polynomial there is at least one local extremum.

Even degree ⇒ has an absolute extremum. Positive leading coefficient: absolute min (opens up). Negative: absolute max (opens down).

Odd degree ⇒ no absolute max or min (on all real numbers).

Degree n has at most n − 1 local extrema and at most n − 2 points of inflection. AP Precalculus Unit 1 · 1.4 1 of 2

Point of inflection: the rate of change switches between increasing and decreasing, so concavity changes.

Guaranteed extrema? f(−2) = 0, f(0) = 5, f(4) = 0. Two zeros with a positive value between them: yes, a local max on −2 < x < 4.

Absolute extremum? f(x) = −3x⁴ + 2x − 5: even, negative LC ⇒ absolute maximum. h(x) = x³ − x: odd ⇒ neither.

Traps. (1) Even degree does not mean a local extremum is absolute; the rule only says one absolute extremum exists. (2) Odd degree has no absolute extrema, but can still have local ones. (3) Give x- values for where, y-values for how big.

Calculator pattern

"When does the number of people change from increasing to decreasing?" Graph the model on the given interval and find the local maximum; report the input (t) value.

Quick check

1. Degree and LC of 3x² + 10 − 8x⁴? 4 and −8

2. Most local extrema a degree-6 polynomial can have? 5

3. Even degree, positive LC: absolute max or min? Min

AP Precalculus Unit 1 · 1.4 2 of 2