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 section is paired with assignments and assessments to reinforce learning.

Week 1: Fundamental Equations and Complex Numbers

Linear and Rational Equations

• Linear Equations: Equations of the form $ax + b = 0$ where $a \neq 0$.

• Rational Equations: Equations involving rational expressions, typically solved by finding a common denominator.

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

Models and Applications

• Applying algebraic equations to real-world scenarios, such as distance, rate, and time problems.

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

Complex Numbers

• Definition: Numbers of the form $a + bi$, where $i = \sqrt{-1}$.

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

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

Week 2: Quadratic Equations and Inequalities

Quadratic Equations

• Standard Form: $ax^2 + bx + c = 0$.

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

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

Other Types of Equations

• Includes radical, absolute value, and higher-degree equations.

• Example: Solve $\sqrt{x + 2} = 5$.

Linear and Absolute Value Inequalities

• Linear Inequalities: Expressions like $ax + b > c$.

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

• Example: Solve $|x - 3| \leq 5$.

Week 3: Matrices and Functions

Matrix Solutions to Linear Systems

• Matrix Representation: Systems of equations can be written as $AX = B$.

• Solution Methods: Row reduction, inverse matrices.

• Example: Solve $\begin{cases} x + y = 2 \\ x - y = 0 \end{cases}$ using matrices.

More on Functions and Their Graphs

• Definition: A function is a relation where each input has exactly one output.

• Graphing: Identifying domain, range, intercepts, and symmetry.

• Example: Graph $f(x) = x^2$.

Week 4: Transformations, Composite, and Inverse Functions

Transformations of Functions

• Shifts, reflections, stretches, and compressions of function graphs.

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

Composite Functions

• Definition: $(f \circ g)(x) = f(g(x))$.

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

Inverse Functions

• Definition: $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$.

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

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

Week 5: Quadratic and Polynomial Functions

Quadratic Functions

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

• Graphing and identifying vertex, axis of symmetry, and direction of opening.

Polynomial Functions and Their Graphs

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

• End behavior, intercepts, and turning points.

Week 6: Polynomial Division, Zeros, and Rational Functions

Dividing Polynomials

• Long division and synthetic division methods.

• Example: Divide $x^3 - 2x^2 + 4$ by $x - 1$.

Zeros of Polynomial Functions

• Finding roots using factoring, Rational Root Theorem, and synthetic division.

• Multiplicity of zeros and their effect on the graph.

Rational Functions and Their Graphs

• Functions of the form $f(x) = \frac{P(x)}{Q(x)}$.

• Asymptotes, holes, and intercepts.

Week 7: Polynomial and Rational Inequalities, Partial Fractions

Polynomial Inequalities

• Solving inequalities such as $P(x) > 0$ by finding zeros and testing intervals.

Rational Inequalities

• Solving $\frac{P(x)}{Q(x)} > 0$ by finding critical points and analyzing sign charts.

Partial Fractions

• Expressing rational expressions as sums of simpler fractions.

• Example: $\frac{5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$.

Week 8: Exponential and Logarithmic Functions

Exponential Functions

• Functions of the form $f(x) = a \cdot b^x$.

• Applications in growth and decay.

Logarithmic Functions

• Inverse of exponential functions: $y = \log_b x$ iff $b^y = x$.

• Properties and graphs of logarithms.

Week 9: Properties and Equations of Exponentials and Logarithms

Properties of Logarithms

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^k) = k \log_b M$

Exponential and Logarithmic Equations

• Solving equations using properties of exponents and logarithms.

• Example: Solve $2^x = 8$ or $\log_2 x = 3$.

Week 10: Review and Final Exam

• Comprehensive review of all topics.

• Practice problems and strategies for the final exam.

Assessment and Assignments

• Weekly reading quizzes and homework assignments for each section.

• Unit tests after major topic clusters.

• Final exam at the end of the course.

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