Precalculus Study Guide: Functions, Graphs, and Equations

Course Overview

Introduction to Precalculus

Precalculus is a foundational mathematics course designed to prepare students for calculus and other advanced mathematical studies. The course focuses on the analysis and interpretation of functions, equations, and their graphs, with applications in modeling and problem-solving.

• Course Format: Online/Hybrid

• Class Schedule: Monday/Wednesday, 7:00 – 9:15 PM

• Credit Hours: 4.5

• Instructor: Jana Reimer

Course Description

Key Concepts in Precalculus

This course emphasizes the analysis and interpretation of the behavior and nature of functions, including polynomial, rational, exponential, and logarithmic functions. Students will learn to solve equations, model real-world scenarios, and apply mathematical reasoning to various types of problems.

• Functions: Polynomial, rational, exponential, logarithmic, and piecewise-defined functions.

• Equations: Systems of equations, modeling, and solving.

• Additional Topics: May include matrices, combinations, sequences and series, and conics.

Course Topics

Chapter Breakdown

The following chapters outline the main topics covered in the course:

• Chapter 1: Graphs and their properties

• Chapter 2: Graphs and their transformations

• Chapter 3: Polynomial and rational functions

• Chapter 4: Quadratic functions

• Chapter 5: Systems of equations and inequalities

• Chapter 6: Exponential and logarithmic functions

• Chapter 12: Sequences and series

Major Topics and Subtopics

Functions and Their Graphs

Functions are mathematical relationships that assign each input exactly one output. Understanding their graphs is essential for analyzing their behavior.

• Definition: A function f is a rule that assigns to each element x in a set called the domain exactly one element f(x) in a set called the range.

• Graphing: The graph of a function is a visual representation of all ordered pairs (x, f(x)).

• Transformations: Shifting, stretching, compressing, and reflecting graphs.

• Example: The graph of $f(x) = x^2$ is a parabola opening upward.

Polynomial and Rational Functions

Polynomial functions are sums of powers of x with coefficients, while rational functions are ratios of polynomials.

• Polynomial Function: $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$

• Rational Function: $f(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials and $Q(x) \neq 0$

• Key Properties: Zeros, end behavior, asymptotes (for rational functions)

• Example: $f(x) = \frac{x^2 - 1}{x - 1}$

Quadratic Functions

Quadratic functions are polynomials of degree two and have parabolic graphs.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$

• Factoring and Solving: Methods include factoring, completing the square, and the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $x^2 - 5x + 6 = 0$

Systems of Equations and Inequalities

Systems involve solving for multiple variables using multiple equations or inequalities.

• Linear Systems: Can be solved by substitution, elimination, or graphing.

• Matrix Methods: Use matrices to represent and solve systems.

• Inequalities: Solutions are often represented as regions on a graph.

• Example: Solve the system: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$

Exponential and Logarithmic Functions

Exponential functions have variables in the exponent, while logarithmic functions are their inverses.

• Exponential Function: $f(x) = a^x$, where $a > 0$ and $a \neq 1$

• Logarithmic Function: $f(x) = \log_a x$, the inverse of $f(x) = a^x$

• Properties: Laws of exponents and logarithms

• Example: $\log_2 8 = 3$ because $2^3 = 8$

Sequences and Series

Sequences are ordered lists of numbers, and series are sums of sequences.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Series: The sum of the terms of a sequence

• Example: Find the sum of the first 5 terms of the sequence $2, 4, 6, 8, 10$

Course Competencies

Skills Developed in Precalculus

Upon completion of this course, students will be able to:

• Analyze and interpret the behavior of various types of functions

• Solve equations and systems of equations

• Model real-world scenarios using mathematical functions

• Apply mathematical reasoning to problem-solving