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, focusing on mastery of fundamental concepts and the use of digital technology in mathematics.

Goals and Student Learning Outcomes

Goal 1: Understanding the Concept of a Function

Students will develop a comprehensive understanding of functions, their properties, and their representations.

• Relation and Function: A relation is a set of ordered pairs, while 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: Function notation, such as f(x), is used to evaluate functions at specific values.

• Intercepts and Zeros: The x-intercept is where the graph crosses the x-axis; the zero of a function is a value where $f(x) = 0$.

• Even and Odd Functions: An even function satisfies $f(-x) = f(x)$; an odd function satisfies $f(-x) = -f(x)$.

• Example: For $f(x) = x^2$, the function is even because $f(-x) = (-x)^2 = x^2 = f(x)$.

Goal 2: Building New Functions from Existing Functions

Students will learn to create new functions through transformations, arithmetic operations, compositions, and inverses.

• Transformations: Includes shifting, stretching, compressing, and reflecting graphs. For example, $f(x) + c$ shifts the graph vertically.

• Combining Functions: Functions can be added, subtracted, multiplied, divided, or composed. For example, $(f \circ g)(x) = f(g(x))$.

• Inverse Functions: The inverse of a function reverses the input and output. If $f(x)$ and $f^{-1}(x)$ are inverses, then $f(f^{-1}(x)) = x$.

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

Goal 3: Building and Analyzing Graphs of Equations and Functions

Students will explore the relationship between equations, tables, and graphs, and learn to construct and analyze graphs using key features.

• Equation, Table, and Graph: Understanding how each representation relates to the others.

• Key Features: Includes intercepts, maxima, minima, intervals of increase/decrease, and asymptotes.

• Analyzing Graphs: Determining domain, range, and other properties from the graph.

• Example: The graph of $f(x) = x^2$ is a parabola opening upward with vertex at (0,0).

Goal 4: Solving Equations, Inequalities, and Systems

Students will apply algebraic and graphical methods to solve equations, inequalities, and systems of equations.

• Solving Equations Algebraically: Techniques include factoring, using the quadratic formula, and isolating variables.

• Solving with Graphs: Solutions correspond to points where the graph intersects the x-axis or other relevant lines.

• Solving Inequalities: Algebraic methods include testing intervals; graphical methods involve shading regions.

• Systems of Linear Equations: Methods include substitution, elimination, and matrix techniques.

• Example: Solve $x^2 - 4 = 0$ algebraically: $x^2 - 4 = 0 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$.

Goal 5: Mastery of Algebraic Techniques for Problem-Solving and Modeling

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

• Interpreting Functions in Context: Understanding how functions model real-world situations.

• Equation of a Line: The equation $y = mx + b$ represents a line with slope $m$ and y-intercept $b$.

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

• Operations and Compositions: Applying function operations to solve application problems.

• Systems of Equations: Modeling situations with two unknowns using systems.

• Example: Population growth can be modeled by $P(t) = P_0 e^{rt}$, where $P_0$ is initial population, $r$ is growth rate, and $t$ is time.

Goal 6: Using Digital Technology in Mathematics

Students will utilize technology to evaluate expressions, sketch graphs, and solve equations and inequalities.

• Exact and Decimal Approximations: Technology can compute values such as $\sqrt{2}$ or $\pi$ to decimal form.

• Evaluating Formulas: Calculators and software can evaluate complex expressions and functions.

• Graphing Functions: Technology assists in visualizing functions and their properties.

• Solving Equations and Inequalities: Software can find solutions numerically or graphically.

• Evaluating Logarithms: Technology can compute logarithmic values, e.g., $\log_{10}(100) = 2$.

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

• Example: Using a graphing calculator to find the intersection point of two lines.

Summary Table: Main Course Topics and Associated Learning Outcomes

Course Topic	Key Learning Outcomes
Functions	Definition, domain/range, representations, notation, intercepts, even/odd properties
Function Operations	Transformations, arithmetic operations, composition, inverses
Graphs	Equation-table-graph relationships, constructing/analyzing graphs, key features
Equations & Inequalities	Algebraic and graphical solutions, systems of equations
Applications	Linear, quadratic, exponential, logarithmic functions, systems, modeling
Technology	Evaluating expressions, graphing, solving, matrix operations

Additional info: Expanded definitions, examples, and context were added to clarify each learning outcome and provide a self-contained study guide suitable for exam preparation.