College Algebra Syllabus and Core Concepts Overview

Course Overview

This study guide summarizes the key topics, structure, and learning outcomes for College Algebra (MATH 1314), as outlined in the course syllabus. The course covers foundational algebraic concepts, functions, polynomial and rational functions, exponential and logarithmic functions, and matrices and linear systems. Mastery of these topics is essential for success in college-level mathematics and related fields.

Course Structure and Evaluation

• Homework: 20% (multiple attempts allowed; late penalty applies)

• Quizzes: 20% (multiple attempts allowed; late penalty applies)

• Midterm Exam: 30% (covers Chapters 2 and 3; one attempt at a testing center)

• Final Exam: 30% (comprehensive; covers Chapters 2, 3, 4, and 6; one attempt at a testing center)

Assignments are completed online via MyMathLab. Exams are proctored at approved testing centers. A graphing calculator (TI-83, TI-83+, TI-84, or TI-84+ series) is required.

Major Topics and Chapter Outline

• Chapter 2: Functions and Their Graphs

• Chapter 3: Polynomial and Rational Functions

• Chapter 4: Exponential and Logarithmic Functions

• Chapter 6: Matrices and Linear Systems

Learning Outcomes

Upon successful completion of the course, students will be able to:

• Demonstrate and apply knowledge of properties of functions, including domain, range, operations, compositions, and inverses.

• Recognize and apply polynomial, rational, radical, exponential, and logarithmic functions and solve related equations.

• Apply graphing techniques to various types of functions.

• Evaluate all roots of higher degree polynomial and rational functions.

• Recognize, solve, and apply systems of linear equations using matrices.

Key Content Areas

Functions and Graphs

• Relation: A set of ordered pairs. Each input (x-value) may correspond to one or more outputs (y-values).

• Function: A relation in which each input has exactly one output.

• Domain: The set of all possible input values (x-values) for a function.

• Range: The set of all possible output values (y-values) for a function.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Graphing Functions: Plot points, identify intercepts, and analyze behavior (increasing, decreasing, constant).

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

• Operations on Functions: Addition, subtraction, multiplication, division, and composition of functions.

• Inverse Functions: A function that "undoes" the action of the original function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$.

Example: If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$.

Polynomial and Rational Functions

• Polynomial Function: A function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $n$ is a non-negative integer and coefficients $a_i$ are real numbers.

• End Behavior: The behavior of the graph as $x \to \infty$ or $x \to -\infty$ depends on the leading term.

• Finding Zeros: Use factoring, the quadratic formula, or synthetic division to find roots of polynomials.

• Rational Zero Theorem: Provides a list of possible rational zeros for a polynomial equation.

• Graphing Rational Functions: Identify vertical, horizontal, and oblique asymptotes; plot intercepts and analyze behavior near asymptotes.

Example: For $f(x) = \frac{x^2 - 1}{x - 1}$, simplify to $f(x) = x + 1$ for $x \neq 1$; vertical asymptote at $x = 1$.

Exponential and Logarithmic Functions

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

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

• Properties:

  • $\log_a (xy) = \log_a x + \log_a y$

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

  • $\log_a (x^r) = r \log_a x$

• Solving Equations: Use properties of exponents and logarithms to solve equations involving these functions.

• Exponential Growth and Decay: Modeled by $y = y_0 e^{kt}$, where $k > 0$ for growth and $k < 0$ for decay.

Example: Solve $2^x = 8$. Since $8 = 2^3$, $x = 3$.

Matrices and Linear Systems

• Matrix: A rectangular array of numbers arranged in rows and columns.

• Solving Systems: Use the Gauss-Jordan method or technology to solve systems of linear equations.

• Matrix Operations: Addition, subtraction, scalar multiplication, and matrix multiplication.

• Inverse of a Matrix: For a square matrix $A$, the inverse $A^{-1}$ satisfies $A A^{-1} = I$, where $I$ is the identity matrix.

• Determinant: A scalar value that can be computed from the elements of a square matrix, used to determine invertibility.

• Cramer's Rule: A method for solving systems of linear equations using determinants.

Example: Solve the system $\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$ using matrices.

Course Policies and Resources

• Attendance: Regular engagement in MyMathLab is required to avoid being dropped from the course.

• Academic Integrity: Cheating, plagiarism, and disruptive behavior are strictly prohibited and subject to disciplinary action.

• Calculator Policy: Only approved TI-83/84 models are allowed during exams; no programmable or non-approved calculators.

• Support: Math Tutoring Lab, instructor office hours, and online resources are available for assistance.

• Accommodations: Students with documented disabilities should contact the Office of Student Accommodations.

Summary Table: Major Topics and Skills

Topic	Key Skills
Functions & Graphs	Define, evaluate, graph, transform, compose, and invert functions; determine domain and range
Polynomial & Rational Functions	Find zeros, analyze end behavior, apply Rational Zero Theorem, graph rational functions, identify asymptotes
Exponential & Logarithmic Functions	Define, graph, apply properties, solve equations, model growth and decay
Matrices & Linear Systems	Define matrices, perform operations, solve systems, find inverses and determinants, apply Cramer's Rule

Additional info: Students are expected to have proficiency in factoring, rational expressions, and operations with exponents and radicals. Remedial coursework is recommended if these skills are lacking.