MTH161 Precalculus Course Schedule: Key Topics and Study Guide

Course Overview

This study guide summarizes the main topics and sections covered in the MTH161 Precalculus course, as outlined in the provided course schedule. The course is structured to build foundational algebraic skills, explore functions and their properties, and introduce advanced topics such as matrices, polynomial and rational functions, and exponential and logarithmic functions. Each week focuses on specific sections, with associated assignments and assessments.

Prerequisites: Fundamental Concepts of Algebra

Linear and Rational Equations

Linear and rational equations form the basis for solving algebraic problems and modeling real-world scenarios.

• Linear Equations: Equations of the form $ax + b = 0$, where a and b are constants.

• Rational Equations: Equations involving rational expressions, such as $\frac{1}{x} + 2 = 3$.

• Solving Techniques: Isolate the variable, clear denominators, and check for extraneous solutions.

• Example: Solve $\frac{2}{x} + 3 = 5$.

Models and Applications

Mathematical models use equations to represent real-world phenomena.

• Application: Setting up equations from word problems.

• Example: If a car travels at 60 mph for t hours, the distance is $d = 60t$.

Complex Numbers

Complex numbers extend the real numbers to include solutions to equations like $x^2 + 1 = 0$.

• Definition: $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$).

• Operations: Addition, subtraction, multiplication, and division of complex numbers.

• Example: $(2 + 3i) + (1 - 4i) = 3 - i$.

Quadratic Equations

Quadratic equations are second-degree equations of the form $ax^2 + bx + c = 0$.

• Solution Methods: Factoring, completing the square, quadratic formula.

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

• Example: Solve $x^2 - 5x + 6 = 0$.

Other Types of Equations

Includes equations such as radical, absolute value, and higher-degree polynomial equations.

• Radical Equations: Equations involving roots, e.g., $\sqrt{x + 2} = 5$.

• Absolute Value Equations: $|x - 3| = 7$.

Linear and Absolute Value Inequalities

Inequalities express relationships where quantities are not necessarily equal.

• Linear Inequalities: $ax + b > 0$ or $ax + b < 0$.

• Absolute Value Inequalities: $|x| < a$ or $|x| > a$.

• Solution Sets: Often expressed in interval notation.

Functions and Graphs

More on Functions and Their Graphs

Functions describe relationships between variables. Their graphs provide visual representations.

• Definition: A function $f$ assigns each input $x$ exactly one output $f(x)$.

• Domain and Range: The set of possible inputs and outputs.

• Graphing: Plotting points $(x, f(x))$ in the coordinate plane.

Transformations of Functions

Transformations shift, reflect, stretch, or compress the graph of a function.

• Vertical Shifts: $f(x) + k$ moves the graph up/down.

• Horizontal Shifts: $f(x - h)$ moves the graph right/left.

• Reflections: $-f(x)$ reflects over the x-axis.

• Example: $f(x) = (x - 2)^2 + 3$ is a parabola shifted right 2 and up 3.

Composite Functions

Composite functions combine two functions into one: $(f \circ g)(x) = f(g(x))$.

• Notation: $f(g(x))$ means apply $g$ first, then $f$.

• Example: If $f(x) = x^2$ and $g(x) = x + 1$, then $f(g(x)) = (x + 1)^2$.

Inverse Functions

The inverse of a function "undoes" the action of the function.

• Definition: $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$.

• Finding Inverses: Swap $x$ and $y$ and solve for $y$.

• Example: $f(x) = 2x + 3$ has inverse $f^{-1}(x) = \frac{x - 3}{2}$.

Polynomial and Rational Functions

Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$.

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

• Graph: Parabola opening up if $a > 0$, down if $a < 0$.

Polynomial Functions and Their Graphs

Polynomial functions are sums of terms with non-negative integer exponents.

• General Form: $f(x) = a_nx^n + \ldots + a_1x + a_0$

• End Behavior: Determined by the leading term $a_nx^n$.

Dividing Polynomials

Polynomials can be divided using long division or synthetic division.

• Long Division: Similar to numerical long division.

• Synthetic Division: Shortcut for dividing by linear factors.

Zeros of Polynomial Functions

Zeros are the values of $x$ for which $f(x) = 0$.

• Finding Zeros: Factor or use the Rational Root Theorem.

• Multiplicity: The number of times a zero is repeated.

Rational Functions and Their Graphs

Rational functions are ratios of polynomials.

• Form: $f(x) = \frac{P(x)}{Q(x)}$

• Asymptotes: Vertical (where $Q(x) = 0$), horizontal, or oblique.

Polynomial and Rational Inequalities

Solving inequalities involving polynomials or rational expressions.

• Test Intervals: Use sign charts to determine solution sets.

• Example: Solve $x^2 - 4 > 0$.

Partial Fractions

Expressing rational functions as sums of simpler fractions.

• Decomposition: $\frac{P(x)}{Q(x)} = \frac{A}{x - r} + \frac{B}{x - s}$, etc.

• Application: Useful for integration and solving equations.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Growth and Decay: Models population, radioactive decay, etc.

• Example: $f(x) = 2^x$ doubles for each increase in $x$ by 1.

Logarithmic Functions

Logarithms are the inverses of exponential functions.

• Definition: $y = \log_a x$ means $a^y = x$.

• Natural Logarithm: $\ln x = \log_e x$.

Properties of Logarithms

Logarithms have several key properties that simplify calculations.

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$

• Quotient Rule: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• Power Rule: $\log_a (x^r) = r \log_a x$

Exponential and Logarithmic Equations

Solving equations involving exponentials and logarithms often requires applying properties and inverses.

• Example: Solve $2^x = 8$ by rewriting as $x = \log_2 8 = 3$.

• Example: Solve $\ln x = 2$ by exponentiating: $x = e^2$.

Matrices and Determinants

Matrix Solutions to Linear Systems

Matrices provide a systematic way to solve systems of linear equations.

• Matrix Form: $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constants matrix.

• Solution Methods: Gaussian elimination, inverse matrices.

• Example: Solve $\begin{cases} x + y = 2 \\ 2x - y = 1 \end{cases}$ using matrices.

Assessment and Assignments

• Weekly Assignments: Reading quizzes and homework for each section, due Sundays at 11:59pm EST.

• Unit Tests: Scheduled after major topic blocks (Units 1-4).

• Final Exam: Comprehensive, scheduled for the last week.

• Note: No extensions on assignments; schedule is subject to change.

Summary Table: Major Topics by Week

Week	Main Topics
1	Linear & Rational Equations, Models & Applications, Complex Numbers
2	Quadratic Equations, Other Types of Equations, Linear & Absolute Value Inequalities
3	Matrix Solutions to Linear Systems, Functions & Graphs
4	Transformations, Composite & Inverse Functions
5	Quadratic & Polynomial Functions
6	Dividing Polynomials, Zeros, Rational Functions
7	Polynomial & Rational Inequalities, Partial Fractions
8	Exponential & Logarithmic Functions
9	Properties of Logarithms, Exponential & Logarithmic Equations
10	Final Exam Review & Final Exam

Additional info: This guide is based on the course schedule and section titles. For detailed examples and practice problems, refer to the course textbook and assigned homework.