College Algebra Syllabus and Core Concepts: MAC 1105 Study Guide

Course Overview

Introduction to College Algebra (MAC 1105)

This course provides a comprehensive study of algebraic concepts, focusing on problem-solving, critical thinking, and computational proficiency. Students will explore equations, inequalities, functions, and their graphs, with an emphasis on quadratic, exponential, and logarithmic functions. The course is designed to prepare students for further studies in mathematics and related fields.

• Prerequisite: MAT 1033 with a grade of C or better, or appropriate placement test score.

• Textbook: College Algebra Essentials, 6th Edition, by Robert F. Blitzer.

• Online Platform: Pearson MyMathLab (access required).

Core Course Topics and Learning Outcomes

1. Equations and Inequalities

Students will learn to solve various types of equations and inequalities using appropriate algebraic techniques.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$.

• Radical Equations: Equations involving roots, such as $\sqrt{x+2} = 5$.

• Polynomial Equations: Equations involving polynomials of degree greater than two.

• Absolute Value Equations and Inequalities: Equations like $|x-3| = 7$ and inequalities such as $|x+2| < 5$.

• Systems of Linear Equations: Solving systems in two or three variables using elimination and substitution.

• Systems of Nonlinear Inequalities: Solving and graphing inequalities involving quadratic or rational expressions.

• Exponential and Logarithmic Equations: Solving equations such as $2^x = 8$ or $\log(x) = 3$.

Example: Solve the quadratic equation $x^2 - 5x + 6 = 0$.

Solution: Factor to $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

2. Functions and Their Properties

Students will define, describe, and analyze functions, including their domains, ranges, and graphs.

• Relations and Functions: A function is a relation in which each input has exactly one output.

• Domain and Range: The domain is the set of all possible input values; the range is the set of all possible output values.

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

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$, used to analyze rates of change.

• Linear Functions: Functions of the form $f(x) = mx + b$.

• Graphing Functions: Identifying key features such as intercepts, symmetry, and intervals of increase/decrease.

• Quadratic, Polynomial, Rational, Exponential, and Logarithmic Functions: Understanding their unique properties and graphs.

Example: Find the domain of $f(x) = \frac{1}{x-2}$.

Solution: The domain is all real numbers except $x = 2$.

3. Manipulating Functions

Students will perform operations on functions, including composition and finding inverses, and use properties of exponents and logarithms.

• Operations with Functions: Addition, subtraction, multiplication, division, and composition: $(f \circ g)(x) = f(g(x))$.

• Inverse Functions: A function $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$.

• Exponential and Logarithmic Forms: $y = \log_b x \iff b^y = x$.

• Evaluating Expressions: Calculating values of exponential and logarithmic expressions.

• Properties of Logarithms:

  • $\log_b(MN) = \log_b M + \log_b N$

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

  • $\log_b(M^k) = k \log_b M$

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

Solution: $(f \circ g)(x) = f(g(x)) = 2x^2 + 1$.

4. Transformations and Graphing

Students will use transformations to write equations for functions and graph them.

• Basic Graphs: Recognize and graph parent functions such as $y = x$, $y = x^2$, $y = \sqrt{x}$, $y = |x|$, $y = 2^x$, $y = \log x$.

• Transformations: Shifts, reflections, stretches, and compressions.

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

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

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

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

Example: Graph $f(x) = (x-2)^2 + 3$.

Solution: This is a parabola shifted right 2 units and up 3 units.

5. Applications and Modeling

Students will apply algebraic concepts to solve real-world problems involving optimization, growth and decay, and systems of equations.

• Optimization with Quadratic Functions: Finding maximum or minimum values, such as maximizing area or profit.

• Rational Function Applications: Solving problems involving rates, mixtures, or proportions.

• Exponential Growth and Decay: Modeling population growth, radioactive decay, or compound interest.

  • General Formula: $A = A_0 e^{kt}$, where $k$ is the growth/decay rate.

• Systems of Equations: Solving application problems involving multiple variables and constraints.

Example: The population of a city grows according to $P(t) = 100,000 e^{0.02t}$, where $t$ is in years. Find the population after 10 years.

Solution: $P(10) = 100,000 e^{0.2} \approx 122,140$.

Course Structure and Assessment

Assessment Components

Category	% of Grade	Details
Homework/Discussions	15%	Unlimited attempts, approx. 0.2% each
Quizzes	10%	3 attempts, timed, approx. 0.6% each
Proctored Chapter Tests	15%	3 attempts, timed, open notes, 5 tests, 3% each
Proctored Exams	60%	1 attempt, timed, NO notes, 4 exams, 15% each

Course Percentage Formula:

$\text{Course percentage} = (\text{homework average})(.15) + (\text{quiz average})(.10) + (\text{chapter tests average})(.15) + (\text{proctored exam #1})(.15) + (\text{proctored exam #2})(.15) + (\text{proctored exam #3})(.15) + (\text{proctored exam #4})(.15)$

Grading Scale:

Grade	Percentage
A	90-100
B	80-89
C	70-79
D	60-69
F	Below 60

Course Schedule (Key Topics by Week)

Week	Main Topics
1	Quadratic Equations, Other Types of Equations
2	Linear and Absolute Inequalities
3-4	Functions and Graphs, Linear Functions, Transformations, Function Operations, Inverse Functions
6-8	Exponential and Logarithmic Functions, Properties, Equations, Growth and Decay
9-10	Quadratic, Polynomial, Rational Functions, Systems of Equations and Inequalities

Additional Information

• Calculator Policy: Only non-graphing calculators without algebraic solvers are permitted (e.g., TI-30Xa, TI-30XIIs, Casio fx-260, Casio fx-300MS Plus).

• Academic Integrity: All work must be your own. Unauthorized use of AI or other resources may result in academic penalties.

• Support Services: Tutoring, technical support, and accommodations are available through HCC and Pearson MyMathLab.

• Communication: Check HCC email and Canvas daily for announcements and updates.

Note: For detailed weekly objectives, assignments, and due dates, refer to the course schedule in Canvas and MyMathLab.