Study Guide: Functions, Graphs, and Polynomial Functions (College Algebra)

Unit Overview: Functions and Polynomial Functions

Introduction

This unit covers key topics from College Algebra, focusing on Functions and Graphs and Polynomial Functions. The schedule includes assigned readings, video lectures, practice problems, and recommended study strategies to help students master these foundational concepts.

Functions and Graphs

Definition and Properties of Functions

A function is a relation that assigns each element in the domain to exactly one element in the range. Functions are fundamental in algebra and are used to model relationships between quantities.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Function Notation: $f(x)$ represents the output of function f for input x.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Example: The function $f(x) = 2x + 3$ maps each real number x to a unique value.

Graphing Functions

Graphing functions helps visualize their behavior and identify key features such as intercepts, maxima, minima, and asymptotes.

• x-intercept: The point(s) where the graph crosses the x-axis ($f(x) = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• Symmetry: Functions may be even (symmetric about the y-axis) or odd (symmetric about the origin).

Example: The graph of $f(x) = x^2$ is a parabola opening upward, symmetric about the y-axis.

Polynomial Functions

Definition and Classification

A polynomial function is a function 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.

• Degree: The highest power of x in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Constant Term: The term without x ($a_0$).

Example: $f(x) = 3x^4 - 2x^2 + 7$ is a fourth-degree polynomial.

Graphing Polynomial Functions

Polynomial functions have smooth, continuous graphs. The degree and leading coefficient determine the end behavior.

• End Behavior: For $f(x) = ax^n$:

  • If n is even and a > 0, both ends rise.

  • If n is even and a < 0, both ends fall.

  • If n is odd and a > 0, left end falls, right end rises.

  • If n is odd and a < 0, left end rises, right end falls.

• Zeros: The x-values where $f(x) = 0$ (roots of the polynomial).

• Multiplicity: The number of times a root is repeated affects how the graph touches or crosses the x-axis.

Example: The graph of $f(x) = x^3 - x$ has zeros at $x = -1, 0, 1$ and shows odd-degree end behavior.

Recommended Study Strategies

Practice and Review

• Use the STUDY PLAN in MyLab to work on practice problems for each section.

• Complete assigned readings and watch video lectures for each topic.

• Work on odd-numbered problems at the end of each textbook section and check answers in the back of the book.

• Review your exam by correcting missed problems and understanding the solutions.

Exam Review and Correction Process

After receiving your exam, use it as a learning tool:

• Review problems you answered correctly to reinforce your understanding.

• Redo missed problems, seeking help if needed, but ensure you can solve them independently.

• Document corrections on a separate sheet, attach to your exam, and organize in your notebook for future reference.

This process helps retain information and improves performance on future assessments.

Unit Schedule Table

The following table summarizes the schedule for readings, video lectures, and homework due dates for Chapters 2 and 3:

Date	Prep (Brightspace)	Section	Read	MyLab Homework Due
Thu 9/25	Video (7:24), Read p. 240-245	2.2	p. 237-250	Sun 10/5
Thu 10/2	Video (4:15), Read p. 285	2.5	p. 284-296	Sun 10/12
Thu 10/9	Video 1 (3:39), Video 2 (2:23)	2.6	p. 300-311	Sun 10/19
Thu 10/16	Video (5:01), Video (5:01)	2.7, 2.8	p. 315-323, p. 326-333	Wed 10/22
Tue 10/21	Video (6:10)	3.1	p. 347-361	Sun 10/26
Thu 10/23	Video (4:39), Read p. 366-371	3.2	p. 366-379	Wed 10/29
Tue 10/28	Video (5:15), Video (3:41)	3.3, 3.4	p. 364-393, p. 397-404	Sun 11/2
Thu 10/30	Video (5:29), Video (6:33)	3.5, 3.6	p. 413-427, p. 433-442	Sun 11/9

Additional info: The table above is reconstructed from the syllabus image and may omit some minor details for clarity.