College Algebra: Course Overview and Study Guide

Course Overview

Introduction to College Algebra

College Algebra is a foundational mathematics course that focuses on the study and analysis of functions, equations, and inequalities. The course prepares students for further study in mathematics and related fields by developing algebraic reasoning and problem-solving skills.

• Instructor: Dr. Isaac Bancroft

• Course Code: MATH 1710

• Meeting Times: MWF 8:00 AM - 8:55 AM and MWF 9:10 AM - 10:05 AM

• Location: Dunn 211

• Textbook: College Algebra - University of Memphis Custom Edition

Main Topics in College Algebra

Functions and Their Properties

Functions are central to algebra and are used to model relationships between quantities. This course covers various types of functions and their characteristics.

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

• Types of Functions: Linear, quadratic, polynomial, root, rational, exponential, and logarithmic functions.

• Key Properties: Domain, range, intercepts, symmetry, and asymptotes.

• Example: The quadratic function $f(x) = ax^2 + bx + c$ is a parabola opening upwards if $a > 0$ and downwards if $a < 0$.

Graphing and Analysis

Graphing functions helps visualize their behavior and identify key features such as intercepts and asymptotes.

• Graphing Techniques: Plotting points, identifying intercepts, and analyzing end behavior.

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

• Example: The graph of $f(x) = (x - h)^2 + k$ is a parabola shifted $h$ units right and $k$ units up from the origin.

Equations and Inequalities

Solving equations and inequalities is a fundamental skill in algebra, involving various algebraic techniques.

• Types: Linear, quadratic, polynomial, rational, exponential, and logarithmic equations and inequalities.

• Methods: Factoring, completing the square, quadratic formula, and properties of exponents and logarithms.

• Quadratic Formula:

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

• Example: To solve $2x^2 - 4x - 6 = 0$, use the quadratic formula with $a = 2$, $b = -4$, $c = -6$.

Systems of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Methods: Substitution, elimination, and matrix methods.

• Example: Solve the system:

  • $x + y = 5$

  • $x - y = 1$

  Solution: Add the equations to get $2x = 6 \Rightarrow x = 3$, then $y = 2$.

Inequalities and Absolute Value

Inequalities express relationships where two expressions are not necessarily equal. Absolute value equations and inequalities require special consideration.

• Solving Linear Inequalities: Use similar steps as solving equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• Absolute Value Equations: $|x| = a$ has solutions $x = a$ or $x = -a$.

• Example: Solve $|2x - 3| = 5$:

$2x - 3 = 5 \Rightarrow x = 4$ $2x - 3 = -5 \Rightarrow x = -1$

Polynomials and Factoring

Polynomials are algebraic expressions with multiple terms. Factoring is the process of expressing a polynomial as a product of simpler polynomials.

• Factoring Techniques: Greatest common factor, grouping, difference of squares, and trinomials.

• Example: Factor $x^2 - 9 = (x - 3)(x + 3)$.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are used to model growth and decay, and to solve equations involving exponents and logarithms.

• Exponential Function: $f(x) = a \cdot b^x$

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

• Properties: $\log_b(xy) = \log_b(x) + \log_b(y)$

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

Course Structure and Grading

Assessment and Grading Policy

The course uses a combination of homework, quizzes, attendance, midterm exams, and a final exam to assess student understanding.

Category	Weight
Homework	10%
Quizzes	10%
Attendance	10%
Midterm Exams	60%
Final Exam	10%

Grades are calculated using the formula:

$\text{Grade} = 0.1 \times (\text{HW AVG}) + 0.1 \times (\text{QUIZ AVG}) + 0.6 \times (\text{MIDTERM AVG}) + 0.1 \times (\text{FINAL})$

Class Structure and Expectations

• Class Format: Traditional lecture with group activities and quizzes.

• Calculator Policy: TI-83, TI-84, TI-89, and certain CASIO calculators are allowed. No cell phones or smartwatches during exams or quizzes.

• Academic Honesty: Cheating is not tolerated. All work must be your own.

• Attendance: Regular attendance is required and contributes to your grade.

• Make-Up Policy: Make-up tests are not offered except for documented emergencies.

Study and Success Tips

• Practice solving a variety of equations and inequalities.

• Work on graphing different types of functions by hand and using technology.

• Review class notes and textbook examples regularly.

• Utilize tutoring resources and office hours for additional help.

Course Schedule Overview

The course is organized into weekly topics, including introductions, function analysis, equations, inequalities, and preparation for exams. Key dates include test days, review sessions, and the final exam.

Sample Weekly Breakdown

• Weeks 1-2: Introduction, Syllabus, Functions (R.1-R.4)

• Weeks 3-4: Review and Test 1, Functions (R.5-R.7, 1.1-1.2)

• Weeks 5-6: Test 2, Functions (1.3-1.6, 2.1-2.2)

• Weeks 7-8: Test 3, Equations (2.3-2.5, Chapter 1-2)

• Weeks 9-10: Test 4, Polynomials (3.1-3.4)

• Weeks 11-12: Test 5, Rational Functions (4.1-4.5)

• Weeks 13-14: Test 6, Exponential and Logarithmic Functions (5.1-5.6)

• Week 15: Final Exam Review and Final Exam

Additional info: The above schedule and content are inferred from the syllabus and may be adjusted by the instructor as needed. Students are encouraged to consult the official course materials for detailed weekly assignments and topics.