Polynomial Functions Overview

Overview

This lecture covers polynomial functions, focusing on definitions, properties, extrema (maximums and minimums), even and odd degree behaviors, and inflection points.

Polynomial Functions and Standard Form

• A polynomial function has the form ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ).
• "Poly" means "many"; polynomials have multiple terms.
• Exponents in standard form decrease from left to right.
• The leading term is the term with the highest exponent.
• The degree of a polynomial is the highest exponent present.
• The leading coefficient is the number in front of the leading term.

Rewriting in Standard Form

• Polynomials should be arranged in descending order of exponents.
• The leading term and coefficient must be taken from standard form, not original order.

Extrema: Maximums and Minimums

• Local (relative) maximum: point where function changes from increasing to decreasing.
• Local (relative) minimum: point where function changes from decreasing to increasing.
• All polynomial functions are continuous with smooth curves.
• Endpoints can be considered local extrema if they are the highest/lowest in their area.

Global (Absolute) Extrema

• The absolute maximum is the highest value over the entire function.
• The absolute minimum is the lowest value over the entire function.
• Arrows on graph indicate the function continues to infinity, affecting existence of absolute extrema.
• Local extrema are not always absolute extrema.

Extrema Between Zeros

• If a polynomial has two zeros, there must be at least one local extremum between them.
• The exact location or type of extremum may require additional information.

Even and Odd Degree Polynomials

• Even-degree polynomials (e.g., ( x^2, x^4 )) have either an absolute minimum (if leading coefficient is positive) or maximum (if negative).
• Odd-degree polynomials (e.g., ( x^3 )) do not have absolute extrema.

Inflection Points and Concavity

• An inflection point is where the rate of change shifts from increasing to decreasing or vice versa.
• Concave up ("like a cup") and concave down ("like a frown") describe the curve's direction.
• A change between concave up and concave down marks an inflection point.

Key Terms & Definitions

• Polynomial — A function consisting of terms with whole number exponents.
• Degree — The highest exponent in a polynomial.
• Leading Coefficient — The coefficient of the term with the highest exponent.
• Standard Form — Arrangement from highest to lowest exponent.
• Local (Relative) Maximum/Minimum — The highest/lowest value in a local region.
• Global (Absolute) Maximum/Minimum — The overall highest/lowest value on the entire function.
• Zero (Root) — Value where the function equals zero.
• Inflection Point — A point where the graph changes concavity.

Action Items / Next Steps

• Rewrite assigned polynomials in standard form.
• Identify degree and leading coefficient for given polynomials.
• Practice finding local and absolute extrema on polynomial graphs.
• Review concavity and identify inflection points on sample functions.