College Algebra (MATH-120) Syllabus and Study Guide

Course Overview

This course in College Algebra (MATH-120) introduces students to the study of families of functions and their graphs, with applications in finance, business, and life, social, and physical sciences. The curriculum covers relations and functions, properties and graphs of linear, quadratic, polynomial, rational, exponential, and logarithmic functions, algebra of functions, solving equations and inequalities, systems of linear equations, and sequences and series.

Course Description and Objectives

• Course Credits: 4 credits

• Prerequisites: Minimum 2.0 in MATH 109 or equivalent placement. Additional placement scores in reading and writing may be required.

• Required Materials: Graphing calculator from the TI-83/84 series (TI-84 Plus CE recommended).

Learning Outcomes

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

1. Determine whether a relation represents a function in numerical, graphical, verbal, and symbolic forms.

2. Use function notation to find and interpret function values.

3. Identify key features of functions, including domain, range, intercepts, intervals of increase/decrease, and end behavior.

4. Perform algebraic operations on functions, including addition, subtraction, multiplication, division, composition, and difference quotient.

5. Recognize and write functions and their inverses numerically, graphically, and symbolically.

6. Construct linear equations from data or verbal descriptions, compute slope, and use linear equations to solve application problems involving arithmetic sequences.

7. Identify and graph quadratic functions to solve maximum or minimum value problems.

8. Construct, solve, and interpret various equations, including polynomial, rational, exponential, logarithmic, complex, linear, and compound inequalities.

9. Graph polynomial, rational, logarithmic, and logistic functions.

10. Solve applications involving exponential growth and decay, and geometric sequences.

Main Topics and Subtopics

1. Relations and Functions

Understanding the foundational concepts of relations and functions is essential in algebra. Functions describe how one quantity depends on another and are represented in various forms.

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

• Representations: Functions can be represented numerically (tables), graphically (graphs), verbally (descriptions), and symbolically (equations).

• Example: The function $f(x) = 2x + 3$ assigns to each $x$ a unique value $f(x)$.

2. Function Notation and Evaluation

Function notation provides a concise way to express the output of a function for a given input.

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

• Evaluation: Substitute the input value into the function's formula to find the output.

• Example: If $f(x) = x^2 - 4x + 7$, then $f(2) = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3$.

3. Key Features of Functions

Analyzing the properties of functions helps in understanding their behavior and applications.

• Domain and Range: The set of all possible input values (domain) and output values (range).

• Intercepts: Points where the graph crosses the axes (x-intercepts and y-intercepts).

• Intervals of Increase/Decrease: Where the function values are rising or falling.

• End Behavior: The behavior of the function as $x$ approaches infinity or negative infinity.

• Example: For $f(x) = x^2$, the domain is all real numbers, the range is $[0, \\infty)$, and the function decreases for $x < 0$ and increases for $x > 0$.

4. Algebra of Functions

Functions can be combined and manipulated using algebraic operations.

• Addition/Subtraction: $(f + g)(x) = f(x) + g(x)$

• Multiplication/Division: $(f \\cdot g)(x) = f(x) \\cdot g(x)$, $(\\frac{f}{g})(x) = \\frac{f(x)}{g(x)}$, $g(x) \\neq 0$

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

• Difference Quotient: $\\frac{f(x+h) - f(x)}{h}$

• Example: If $f(x) = x + 1$ and $g(x) = 2x$, then $(f + g)(x) = x + 1 + 2x = 3x + 1$.

5. Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs.

• Definition: If $f(a) = b$, then $f^{-1}(b) = a$.

• Finding Inverses: Solve $y = f(x)$ for $x$ in terms of $y$, then interchange $x$ and $y$.

• Example: For $f(x) = 3x + 2$, the inverse is $f^{-1}(x) = \\frac{x - 2}{3}$.

6. Linear Equations and Applications

Linear equations model relationships with constant rates of change and are widely used in real-world applications.

• Slope: $m = \\frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a Line: $y = mx + b$

• Arithmetic Sequences: $a_n = a_1 + (n-1)d$

• Example: The line passing through $(1,2)$ and $(3,6)$ has slope $m = \\frac{6-2}{3-1} = 2$ and equation $y = 2x$.

7. Quadratic Functions and Applications

Quadratic functions are polynomials of degree two and are used to model parabolic relationships.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\\frac{b}{2a}$

• Maximum/Minimum: The vertex represents the maximum or minimum value of the function.

• Example: For $f(x) = -2x^2 + 4x + 1$, the vertex is at $x = -\\frac{4}{-4} = 1$.

8. Polynomial, Rational, Exponential, and Logarithmic Functions

These functions extend the concept of algebraic relationships to more complex forms.

• Polynomial: $f(x) = a_nx^n + ... + a_1x + a_0$

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

• Exponential: $f(x) = ab^x$

• Logarithmic: $f(x) = \\log_b(x)$

• Example: $f(x) = 2^x$ is an exponential function; $f(x) = \\log_{10}(x)$ is a logarithmic function.

9. Solving Equations and Inequalities

Solving equations and inequalities is a core skill in algebra, involving various types of expressions.

• Linear Equations: $ax + b = 0$

• Quadratic Equations: $ax^2 + bx + c = 0$

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

• Compound Inequalities: Solve each part separately and find the intersection or union of solutions.

• Example: Solve $2x - 5 > 3$: $2x > 8$, $x > 4$.

10. Applications: Exponential Growth/Decay and Sequences

Exponential and geometric models are used to describe growth, decay, and patterns in sequences.

• Exponential Growth/Decay: $A = A_0e^{kt}$, where $k > 0$ for growth, $k < 0$ for decay.

• Geometric Sequence: $a_n = a_1r^{n-1}$

• Example: If a population grows by 5% per year, $A = A_0(1.05)^t$.

Course Materials

• Graphing Calculator: Required (TI-83/84 series recommended).

Grading and Evaluation

Grades are determined by a combination of participation, exams, final exam, projects, quizzes, and other assignments. The following table summarizes the grading scale:

College Standard Numeric Grade	Narrative Equivalent	Guideline for Performance Achievement of Objectives
4.0	Excellent	91% to 100%
3.5		86% to 90%
3.0	Good	81% to 85%
2.5		76% to 80%
2.0	Satisfactory	71% to 75%
1.5		66% to 70%
1.0	Poor	60% to 65%

Course Policies and Resources

• Artificial Intelligence: Use of AI tools is not allowed unless approved by the instructor for specific assignments.

• Proctored Testing: Some exams may require in-person or approved remote proctoring.

• Participation: Regular engagement and completion of assignments are required.

• Institutional Support: Academic advising, student support resources, and success coaches are available.

• Student Code of Conduct: Students must adhere to college rules and guidelines for academic integrity and conduct.

Contact Hours

Lecture	Lab	Other	Total Hrs
64	0	0	64

Additional Information

• Emergency closure and remote instruction policies are in place for unforeseen events.

• Transfer information and academic support resources are available through the college.

• Nondiscrimination and media release statements apply to all students.