College Algebra (Math 1710) – Course Structure, Topics, and Study Guide

Course Overview

This study guide summarizes the key information and academic content for Math 1710: College Algebra as outlined in the course syllabus. College Algebra is a foundational mathematics course designed to enhance students' algebraic skills, problem-solving abilities, and understanding of mathematical modeling, with applications relevant to both STEM and non-STEM fields.

Course Structure and Requirements

• Prerequisites: Two years of high school algebra and a qualifying ACT or COMPASS placement score.

• Required Materials: E-book "College Algebra with Modeling & Visualization" (6th edition, Rockswold) and a TI-83 or TI-84 Plus graphing calculator.

• Course Format: Online with required in-person, proctored final exam.

• Grading:

  • Online homework: 25% of final grade

  • Five (5) proctored online tests: 60% of final grade (lowest test grade dropped)

  • Comprehensive in-person final exam: required to pass the course

• Attendance: Regular participation and activity completion required; attendance tracked via D2L platform.

Learning Outcomes

Upon successful completion, students will be able to:

• Demonstrate enhanced mathematical and problem-solving skills.

• Apply algebraic methods to solve practical problems.

• Utilize graphing calculators to deepen understanding of algebraic concepts.

• Understand functions from graphical, numeric, and symbolic perspectives.

• Work with polynomial, rational, exponential, and logarithmic functions, including real-world modeling applications.

• Solve systems of linear equations using various methods, including matrix methods.

Course Topics and Sections

The course covers selected topics from the following chapters:

• 1. Introduction to Functions and Graphs (Sections 1.1, 1.2, 1.3, 1.4)

• 2. Linear Functions, Equations, and Lines (Sections 2.1, 2.2, 2.3, 2.4, 2.5)

• 3. Quadratic Functions and Equations (Sections 3.1, 3.2, 3.3, 3.4, 3.5)

• 4. Non-Linear Functions and Equations (Sections 4.1, 4.2, 4.6)

• 5. Exponential and Logarithmic Functions (Sections 5.1, 5.2, 5.3, 5.4, 5.5, 5.6)

• 6. Systems of Equations and Inequalities (Sections 6.1, 6.2, 6.3, 6.4, 6.5, 6.6)

• 8. Further Topics: Counting Principles and Probability (Sections 8.3, 8.6)

Key Topics and Concepts

Functions and Graphs

Functions are fundamental objects in algebra, representing relationships between variables. Understanding their properties and graphical representations is essential.

• Definition: A function is a relation in which each input (domain) has exactly one output (range).

• Notation: $f(x)$ denotes the value of the function $f$ at input $x$.

• Graphing: The graph of a function is the set of all points $(x, f(x))$ in the coordinate plane.

• Example: The function $f(x) = 2x + 3$ is a linear function whose graph is a straight line.

Linear Functions, Equations, and Lines

Linear functions model constant rates of change and are represented by straight lines in the coordinate plane.

• General Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope: Measures the steepness of the line: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Applications: Used to model real-world situations with constant change, such as speed or cost calculations.

• Example: If a taxi charges a $3 base fare plus $2 per mile, the cost function is$C(m) = 2m + 3$.

Quadratic Functions and Equations

Quadratic functions describe parabolic relationships and are essential in modeling acceleration, area, and other phenomena.

• Standard Form: $y = ax^2 + bx + c$

• Vertex: The highest or lowest point of the parabola, given by $x = -\frac{b}{2a}$.

• Factoring, Completing the Square, Quadratic Formula: Methods for solving quadratic equations.

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

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

Non-Linear Functions and Equations

Non-linear functions include polynomial, rational, and other types of functions that do not graph as straight lines.

• Polynomial Functions: Functions of the form $f(x) = a_nx^n + \ldots + a_1x + a_0$.

• Rational Functions: Ratios of polynomials, $f(x) = \frac{P(x)}{Q(x)}$.

• Asymptotes: Lines that the graph approaches but never touches (vertical, horizontal, or slant).

• Example: $f(x) = \frac{1}{x}$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

Exponential and Logarithmic Functions

These functions model growth and decay processes, such as population growth, radioactive decay, and financial interest.

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

• Logarithmic Function: $f(x) = \log_b(x)$, the inverse of the exponential function.

• Properties: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$.

• Example: The population of a bacteria culture doubles every hour: $P(t) = P_0 \cdot 2^t$.

Systems of Equations and Inequalities

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

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

• Matrix Representation: Systems can be written as $A\vec{x} = \vec{b}$, where $A$ is a matrix of coefficients.

• Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ by addition: $2x = 6 \Rightarrow x=3$, $y=2$.

Counting Principles and Probability

These topics introduce basic combinatorics and probability theory, useful for analyzing outcomes and making predictions.

• Counting Principles: Fundamental Counting Principle, permutations, and combinations.

• Probability: The likelihood of an event, $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.

• Example: The number of ways to choose 3 students from 10: $\binom{10}{3} = 120$.

Summary Table: Main Types of Functions Covered

Type of Function	General Form	Graph Shape	Key Properties
Linear	$y = mx + b$	Straight line	Constant rate of change (slope), y-intercept
Quadratic	$y = ax^2 + bx + c$	Parabola	Vertex, axis of symmetry, opens up/down
Polynomial	$y = a_nx^n + \ldots + a_0$	Varies (depends on degree)	End behavior, turning points
Rational	$y = \frac{P(x)}{Q(x)}$	Hyperbola, others	Asymptotes, domain restrictions
Exponential	$y = a \cdot b^x$	Increasing/decreasing curve	Growth/decay, horizontal asymptote
Logarithmic	$y = \log_b(x)$	Increasing curve	Inverse of exponential, vertical asymptote

Academic Integrity and Support

• Academic Honesty: Plagiarism, cheating, and fabrication are strictly prohibited and may result in disciplinary action.

• Support Services: Math tutoring is available to all students; contact the university's tutoring center for details.

• Disability Accommodations: Students requiring accommodations should contact the Disability & Access Center (DAC) in advance.

Important Dates and Policies

• Attendance reporting, drop deadlines, and exam dates are provided in the course schedule and university calendar.

• Final exam must be taken in person at the MTSU Testing Center or approved site; missing the final results in a failing grade.

• No make-up exams except for documented emergencies with instructor approval.

Conclusion

College Algebra (Math 1710) provides essential mathematical skills and concepts for further study and real-world application. Mastery of the topics outlined above will prepare students for success in mathematics and related disciplines.