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

• Instructor: Jana Reimer

• Course Duration: Fall 2025

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 also study absolute value, piecewise-defined functions, systems of equations, modeling, and solving word problems.

• Functions: Understanding different types of functions and their properties.

• Graphs: Learning how to graph functions and interpret their graphical representations.

• Equations: Solving various types of equations, including linear, quadratic, and higher-order equations.

• Modeling: Applying mathematical concepts to real-world scenarios.

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

Main Topics and Chapters

Course Outline

The following chapters outline the main topics covered in the Precalculus 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

• Chapter 13: Conics

Detailed Topic Summaries

Graphs and Their Properties

Graphs are visual representations of mathematical functions and equations. Understanding the properties of graphs is essential for analyzing the behavior of functions.

• Key Point 1: The graph of a function shows the relationship between the independent variable (usually x) and the dependent variable (usually y).

• Key Point 2: Important properties include intercepts, symmetry, asymptotes, and intervals of increase/decrease.

• Example: The graph of $y = x^2$ is a parabola opening upwards with its vertex at the origin.

Polynomial and Rational Functions

Polynomial functions are expressions involving powers of x with real coefficients. Rational functions are ratios of two polynomials.

• Key Point 1: A polynomial function has the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$.

• Key Point 2: Rational functions are of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

• Example: $f(x) = \frac{x^2 - 1}{x + 2}$ is a rational function.

Quadratic Functions

Quadratic functions are a special case of polynomial functions of degree 2. Their graphs are parabolas.

• Key Point 1: The standard form is $f(x) = ax^2 + bx + c$.

• Key Point 2: The vertex of the parabola is at $x = -\frac{b}{2a}$.

• Example: $f(x) = 2x^2 - 4x + 1$ has its vertex at $x = 1$.

Systems of Equations and Inequalities

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

• Key Point 1: Solutions can be found using substitution, elimination, or graphing methods.

• Key Point 2: Systems of inequalities define regions in the plane.

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

Exponential and Logarithmic Functions

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

• Key Point 1: Exponential function: $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Key Point 2: Logarithmic function: $f(x) = \log_a x$, the inverse of $a^x$.

• Example: $f(x) = 2^x$ and $g(x) = \log_2 x$ are inverse functions.

Sequences and Series

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

• Key Point 1: Arithmetic sequence: $a_n = a_1 + (n-1)d$

• Key Point 2: Geometric sequence: $a_n = a_1 r^{n-1}$

• Example: The sequence 2, 4, 6, 8 is arithmetic with $d = 2$.

Conic Sections

Conic sections are curves formed by the intersection of a plane and a cone: circles, ellipses, parabolas, and hyperbolas.

• Key Point 1: Standard equations include:

  • Circle: $(x - h)^2 + (y - k)^2 = r^2$

  • Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

  • Parabola: $y = ax^2 + bx + c$

  • Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

• Key Point 2: Each conic has unique properties and applications in geometry and physics.

• Example: The equation $x^2 + y^2 = 9$ represents a circle with center at (0,0) and radius 3.

Course Competencies

Expected Skills and Outcomes

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

• Analyze and interpret various types of functions and their graphs.

• Solve equations and systems of equations using appropriate methods.

• Apply mathematical concepts to model and solve real-world problems.

• Understand and utilize properties of sequences, series, and conic sections.

*Additional info: Some topics such as matrices, combinations, and advanced modeling may be covered as additional material depending on instructor preference and time available.*