College Algebra Syllabus and Core Competencies – Study Guide

Course Overview

This study guide summarizes the key topics, objectives, and competencies for College Algebra (MA 116) at Washburn University. The course is designed to develop students' quantitative reasoning and mathematical problem-solving skills, focusing on equations, inequalities, functions, graphing, and applications.

Core Course Competencies

I. Analysis and Graphing of Functions and Non-Functions

Understanding and working with functions is central to college algebra. This includes recognizing, analyzing, and graphing various types of functions and relations.

• Function Notation: Use and interpret notation such as $f(x)$ to represent functions.

• Functions vs. Relations: Distinguish between functions (each input has exactly one output) and relations (inputs may have multiple outputs).

• Graphing: Sketch graphs of the following types:

  • Constant

  • Linear

  • Quadratic

  • Absolute Value

  • Piece-wise Defined

  • Square Root

  • Cubic

  • Polynomial

  • Rational

  • Exponential

  • Logarithmic

  • Circle (as a relation)

• Key Graph Features: Identify and use intercepts, asymptotes, domain, range, and transformations (shifts, stretches, reflections).

• Equation Writing: Write equations for the above functions or a circle given a description or graph.

• Function Analysis: Use graphs to analyze function behavior, such as increasing/decreasing intervals and extrema.

• Combining Functions: Find arithmetic combinations (sum, difference, product, quotient) and compositions of functions: $$(f \circ g)(x) = f(g(x))$$

• Inverse Functions: Find and verify the inverse of a function, if it exists.

Example: The function $f(x) = 2x + 3$ is linear. Its graph is a straight line with slope 2 and y-intercept 3. The inverse is $f^{-1}(x) = \frac{x-3}{2}$.

II. Solutions of Equations and Inequalities

Solving equations and inequalities is a foundational skill in algebra, with applications in modeling and problem-solving.

• Types of Equations: Solve equations including:

  • Literal equations (solving for a variable in terms of others)

  • Quadratic equations (by factoring and quadratic formula): $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

  • Rational equations (involving fractions with variables in the denominator)

  • Radical equations (involving roots)

  • Absolute value equations: $$|x| = a \implies x = a \text{ or } x = -a$$

  • Exponential and logarithmic equations

• Types of Inequalities: Solve inequalities such as:

  • Linear (one and two variables)

  • Polynomial

  • Rational

  • Absolute value

• Applications: Apply equations and inequalities to real-world problems, including:

  • Depreciation

  • Growth and decay (exponential models): $$A = A_0 e^{kt}$$

  • Maximum/minimum problems (optimization)

• Data Analysis and Modeling: Examine data, make predictions, and interpret results using mathematical models.

• Systems of Equations: Solve systems using various methods, including substitution, elimination, and matrices.

Example: To solve $2x^2 - 3x - 5 = 0$, use the quadratic formula: $$x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} = \frac{3 \pm \sqrt{49}}{4}$$

Course Objectives

• Solve a variety of mathematical equations and inequalities.

• Develop mathematical functions and/or equations to model a situation.

• Identify characteristics and sketch various types of graphs.

• Analyze a situation and solve problems using a formula or graph.

Assessment and Grading

The course grade is based on the following components:

Component	Percentage
Review Quiz	5%
Hour Exams (4)	55%
MyLabMath Assignments	10%
Assessment Worksheets	8%
Attendance	2%
Cumulative Final Exam	20%
Total	100%

Required Materials

• Textbook: MyLabMath online access to College Algebra: Enhanced with Graphing Utilities (8th edition) by Sullivan and Sullivan.

• Calculator: TI-83 or TI-84 graphing calculator (any version).

• Other: 3-ring binder, notebook paper, dividers.

General Education Outcomes

• Quantitative Reasoning: Ability to work with numerical data and make/understand mathematical arguments.

• Scientific Literacy: Apply mathematical and scientific methods to solve problems in various contexts.

• Communication: Clearly communicate arguments using words, tables, graphs, equations, and functions.

Additional Information

• Attendance: Regular attendance and participation are required for success.

• Academic Integrity: All work must be your own. Use of AI or copying is prohibited on graded assignments.

• Technology: Phones must be put away during class. Calculators are required; phones may not be used as calculators.

• Help: Seek help immediately if you have difficulty understanding material. Use office hours and email for private discussions.

Additional info: This guide is based on the course syllabus and core competencies. For detailed examples and practice problems, refer to the course textbook and MyLabMath assignments.