College Algebra Syllabus and Course Structure – Study Guide

Course Overview

This course in College Algebra (MAC 1105) is designed to develop students' problem-solving skills, critical thinking, computational proficiency, and contextual fluency through the study of equations, functions, and their graphs. The course emphasizes quadratic, exponential, and logarithmic functions, and covers a range of foundational algebraic concepts.

Key Course Information

• Course Title: College Algebra

• Course Number: MAC 1105

• Credit Hours: 3.0

• Prerequisite: MAT1033

• Required Textbook: College Algebra: Graphs and Models (Blitzer, 7th Edition, ISBN: 9780132840622)

• Calculator Policy: Scientific calculators are required. Graphing calculators are not permitted during exams.

Major Topics and Learning Objectives

The following topics outline the main content areas and learning outcomes for the course.

1. Functions and Function Notation

Understanding the concept of a function is central to algebra. Functions describe relationships between variables, typically written as f(x).

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

• Notation: Functions are commonly written as $f(x)$, where $x$ is the input variable.

• Example: $f(x) = 2x + 3$ assigns to each $x$ the value $2x + 3$.

2. Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Finding the Domain: Exclude values that cause division by zero or negative values under an even root.

• Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x = 2$.

3. Graphing Functions and Relations

Graphing is a visual way to represent functions and their properties.

• Key Features: Intercepts, symmetry, asymptotes, and intervals of increase/decrease.

• Example: The graph of $f(x) = x^2$ is a parabola opening upwards.

4. Operations on Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

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

5. Inverse Functions

An inverse function reverses the effect of the original function, denoted $f^{-1}(x)$.

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

• Finding the Inverse: Swap $x$ and $y$ in $y = f(x)$ and solve for $y$.

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

6. Types of Functions

The course covers several important types of functions:

• Linear Functions: $f(x) = mx + b$

• Quadratic Functions: $f(x) = ax^2 + bx + c$

• Rational Functions: $f(x) = \frac{P(x)}{Q(x)}$, $Q(x) \neq 0$

• Absolute Value Functions: $f(x) = |x|$

• Radical Functions: $f(x) = \sqrt[n]{x}$

• Exponential Functions: $f(x) = a^x$, $a > 0$, $a \neq 1$

• Logarithmic Functions: $f(x) = \log_a(x)$, $a > 0$, $a \neq 1$

7. Transformations of Graphs

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

• Vertical Shift: $f(x) + k$ shifts up/down by $k$ units.

• Horizontal Shift: $f(x - h)$ shifts right/left by $h$ units.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

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

8. Solving Equations and Inequalities

Solving equations and inequalities is a fundamental skill in algebra.

• Quadratic Equations: $ax^2 + bx + c = 0$ can be solved by factoring, completing the square, or using the quadratic formula:

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

• Rational Equations: Clear denominators and solve the resulting equation.

• Radical Equations: Isolate the radical and square both sides, checking for extraneous solutions.

• Inequalities: Solve similarly to equations, but remember to reverse the inequality when multiplying/dividing by a negative.

9. Systems of Equations and Applications

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

• Methods: Substitution, elimination, and graphical methods.

• Example: Solve $\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$

• Applications: Word problems involving mixtures, investments, and other real-world scenarios.

Assessment and Grading

• Unit Exams: 54% of final grade (three exams at 18% each)

• Comprehensive Final Exam: 21% of final grade

• Homework: 25% of final grade (completed in MyMathLab)

• Grading Scale:

Grade	Percentage
A	90-100%
B+	87-89%
B	80-86%
C+	77-79%
C	70-76%
D	66-69%
F	Below 60%

Course Policies and Resources

• Attendance: Students are responsible for attending lectures and completing assignments. Attendance is not factored into the course grade but is strongly encouraged.

• Academic Honesty: Cheating and academic dishonesty are not tolerated. Exams are closed-book and closed-notes. Use of unauthorized materials or devices will result in a zero for the exam.

• Homework: Unlimited attempts per problem in MyMathLab. Only problems completed before the due date receive credit.

• Exams: Four required exams (three unit exams, one comprehensive final). No make-up exams except for documented emergencies.

• Calculator Policy: Scientific calculators allowed; graphing calculators not permitted during exams.

• Proctoring: All exams are proctored and given in class.

Course Schedule (Sample)

• August 18 - September 17: Chapters 1 and 6

• September 22: Exam 1

• September 24 - October 8: Chapter 2

• October 13: Exam 2

• October 15 - November 3: Chapter 3

• November 5: Exam 3

• November 12 - December 3: Chapters 4 and 5

• December 8: Comprehensive Final Exam

Institutional Learning Outcomes

• ILO 3.1: Perform accurate computations using order of operations with and without technology.

• ILO 3.2: Identify and organize relevant information and complete the solution of an applied problem.

• ILO 3.3: Interpret and communicate understanding of visual representations of data.

• ILO 3.4: Demonstrate mathematical number sense and unit sense.