Math 1111: College Algebra Course Description, Goals, and Student Learning Outcomes

Course Overview

This course provides an in-depth study of algebraic, exponential, and logarithmic functions, emphasizing both algebraic and graphical techniques for solving a variety of mathematical problems. The curriculum is designed to prepare students for calculus and mathematical modeling by developing mastery of core algebraic concepts and digital technology skills.

Major Topics and Learning Outcomes

Functions

Understanding functions is central to college algebra. Students will learn to define, analyze, and manipulate functions in various forms.

• Relation and Function: A relation is a set of ordered pairs; a function is a relation in which each input corresponds to 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.

• Multiple Representations: Functions can be represented as equations, tables, graphs, or verbal descriptions.

• Function Notation: If $f$ is a function, $f(x)$ denotes the output for input $x$.

• Intercepts and Zeros: The x-intercept is where the graph crosses the x-axis ($f(x)=0$); the y-intercept is where it crosses the y-axis ($x=0$).

• Even and Odd Functions: A function $f$ is even if $f(-x) = f(x)$ for all $x$; odd if $f(-x) = -f(x)$ for all $x$.

Building New Functions

Students will learn to create new functions from existing ones using transformations, arithmetic operations, and compositions.

• Transformations: Includes shifting, stretching, compressing, and reflecting graphs.

• Arithmetic Operations: Functions can be added, subtracted, multiplied, or divided: $$(f+g)(x) = f(x) + g(x)$$(f \cdot g)(x) = f(x) \cdot g(x)$$

• Composition: The composition $f \circ g$ is defined as $$(f \circ g)(x) = f(g(x))$$

• Inverse Functions: The inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Graphs of Equations and Functions

Graphical analysis is a key skill for understanding the behavior of functions and equations.

• Equation, Table, and Graph: Recognize how equations, tables, and graphs represent the same function.

• Constructing Graphs: Use key features such as intercepts, maxima, minima, and asymptotes to sketch graphs.

• Analyzing Graphs: Identify domain, range, intervals of increase/decrease, and symmetry from a graph.

Equations, Inequalities, and Systems

Solving equations and inequalities is fundamental in algebra, both algebraically and graphically.

• Solving Equations Algebraically: Use techniques such as factoring, substitution, and the quadratic formula: $$ax^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Solving Equations Graphically: Find solutions by identifying where the graph crosses the x-axis.

• Solving Inequalities: Algebraic methods include sign charts and interval testing; graphical methods involve analyzing regions above or below the x-axis.

• Systems of Linear Equations: Solve using substitution, elimination, or matrix methods.

Algebraic Techniques and Mathematical Modeling

Students will apply algebraic methods to real-world problems and mathematical modeling.

• Interpreting Functions: Understand the meaning of functions in applied contexts.

• Equations of Lines: The slope-intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Applications: Solve problems involving linear, quadratic, exponential, and logarithmic functions.

• Operations and Compositions: Apply arithmetic and composition to solve application problems.

• Systems Applications: Model and solve systems with two unknowns.

Digital Technology in Mathematics

Technology is integrated throughout the course to enhance understanding and computation.

• Exact and Approximate Values: Use calculators or software to evaluate expressions and approximate irrational numbers.

• Evaluating Functions: Technology can compute function values efficiently.

• Graphing: Use graphing calculators or software to sketch and analyze functions.

• Solving Equations and Inequalities: Technology assists in finding solutions numerically or graphically.

• Evaluating Logarithms: Use technology for logarithmic calculations.

• Matrix Operations: Technology can perform matrix addition, multiplication, and inversion.

Summary Table: Course Goals and Student Learning Outcomes

Goal	Student Learning Outcomes (SLOs)
Understand Functions	Define relation/function, identify domain/range, recognize functions, apply notation, find intercepts/zeros, determine even/odd
Build New Functions	Perform transformations, combine functions, determine inverses
Analyze Graphs	Relate equation/table/graph, construct graphs, analyze key features
Solve Equations/Inequalities/Systems	Solve algebraically and graphically, solve systems
Algebraic Techniques & Modeling	Interpret functions, write equations, solve applications (linear, quadratic, exponential, logarithmic, systems)
Digital Technology	Evaluate expressions, formulas, functions, sketch graphs, solve equations/inequalities, evaluate logarithms, matrix operations