Polynomial Functions: Graphs, Extrema, and Interval Notation (Precalculus Review)

Polynomial Functions and Their Graphs

Introduction

Polynomial functions are a central topic in precalculus, involving expressions of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$, where the coefficients $a_i$ are real numbers and $n$ is a non-negative integer. Understanding their graphs, extrema, and interval notation is essential for analyzing their behavior.

Key Concepts in Polynomial Graphs

• Local Minimum: A point on the graph where the function value is lower than at all nearby points. Formally, $f(x_0)$ is a local minimum if there exists an interval $(x_0 - ext{h}, x_0 + ext{h})$ such that $f(x_0) \\leq f(x)$ for all $x$ in the interval.

• Local Maximum: A point on the graph where the function value is higher than at all nearby points. $f(x_0)$ is a local maximum if $f(x_0) \\geq f(x)$ for all $x$ in a neighborhood of $x_0$.

• Absolute Maximum: The highest value of $f(x)$ over the entire domain.

• Absolute Minimum: The lowest value of $f(x)$ over the entire domain.

• Domain: The set of all possible input values ($x$) for which the function is defined.

• Interval Notation: A concise way to describe subsets of the real number line. For example, $(-5, 0)$ means all real numbers $x$ such that $-5 < x < 0$.

Analyzing Extrema from Graphs

To determine local and absolute extrema from a graph:

1. Identify peaks (local maxima) and valleys (local minima).

2. Compare all function values to find the absolute maximum and minimum.

3. Note the domain restrictions, as extrema may occur at endpoints.

Example: Extrema from Polynomial Graphs

Suppose the graph of $f(x)$ is given for $-6 < x < 6$. To answer questions such as "Where does $f$ have a local minimum?" or "What is the absolute maximum of $f$?", follow these steps:

• Scan the graph for lowest and highest points within the domain.

• Mark $x$-values where the graph changes direction (from decreasing to increasing for minima, and vice versa for maxima).

• Use interval notation to specify the domain and location of extrema.

Interval Notation

• Open Interval: $(a, b)$ includes all $x$ such that $a < x < b$.

• Closed Interval: $[a, b]$ includes all $x$ such that $a \\leq x \\leq b$.

• Half-Open Interval: $[a, b)$ or $(a, b]$ includes $a$ or $b$ respectively.

Sample Table: Types of Extrema

Type	Definition	How to Identify
Local Minimum	Lowest point in a neighborhood	Valley in the graph
Local Maximum	Highest point in a neighborhood	Peak in the graph
Absolute Minimum	Lowest point on the entire domain	Lowest $y$-value in the domain
Absolute Maximum	Highest point on the entire domain	Highest $y$-value in the domain

Worked Example

Problem: Given the graph of $g(x)$ for $-5 < x < 0$, identify the local and absolute extrema.

• Local Minimum: Find $x$ where the graph has a valley.

• Local Maximum: Find $x$ where the graph has a peak.

• Absolute Maximum: Highest point in $-5 < x < 0$.

• Absolute Minimum: Lowest point in $-5 < x < 0$.

Answer: (Values depend on the specific graph; in practice, read $x$-values from the graph.)

Summary of Steps for Extrema Analysis

1. Read the domain from the problem statement.

2. Identify all local minima and maxima by observing changes in direction.

3. Compare all $y$-values to find absolute extrema.

4. Express answers in interval notation where appropriate.

Additional info:

• Questions in the worksheet focus on reading and interpreting polynomial graphs, identifying extrema, and using interval notation.

• These skills are foundational for further study in calculus, where critical points and intervals of increase/decrease are analyzed using derivatives.