College Algebra: Course 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 further mathematical study by developing mastery in function analysis, equation solving, and mathematical modeling.

Major Course Goals and Student Learning Outcomes (SLOs)

Understanding Functions

Students will develop a comprehensive understanding of the concept of a function, including its definition, properties, and representations.

• Definition of Relation and Function: A relation is a set of ordered pairs. A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Domain and Range: The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Recognizing Functions: Functions can be represented as equations, tables, graphs, or verbal descriptions. The vertical line test is used to determine if a graph represents a function.

• Function Notation: Functions are often written as $f(x)$, where $x$ is the input variable. To evaluate, substitute the value into the function.

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

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

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Building New Functions from Existing Functions

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

• Transformations: Shifting, reflecting, stretching, and compressing graphs. For example, $f(x) + c$ shifts up, $f(x - c)$ shifts right.

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

• Inverse Functions: The inverse $f^{-1}(x)$ reverses the effect of $f(x)$. A function has an inverse if it is one-to-one.

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

Graphing and Analyzing Functions

Students will construct and analyze graphs, connecting equations, tables, and graphical representations to understand function behavior.

• Multiple Representations: Functions can be described by equations, tables of values, and graphs.

• Key Features: Intercepts, maxima/minima, intervals of increase/decrease, and asymptotes are important for graph analysis.

• Graph Construction: Use key features and transformations to sketch graphs.

• Example: The graph of $f(x) = |x|$ has a vertex at (0,0) and is symmetric about the y-axis.

Solving Equations, Inequalities, and Systems

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

• Algebraic Solutions: Use algebraic manipulation to solve linear, quadratic, and other types of equations.

• Graphical Solutions: The solution to $f(x) = g(x)$ is the x-value(s) where the graphs of $f$ and $g$ intersect.

• Inequalities: Solve inequalities algebraically and graphically, representing solutions on a number line or coordinate plane.

• Systems of Equations: Solve systems using substitution, elimination, or graphing.

• Example: Solve $2x + 3 = 7$ algebraically: $x = 2$.

Algebraic Techniques and Mathematical Modeling

Students will master algebraic techniques and apply them to real-world problems and mathematical models.

• Interpreting Functions in Context: Understand how functions model real-world phenomena.

• Writing Equations of Lines: The equation of a line in slope-intercept form is $y = mx + b$.

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

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

• Systems Applications: Use systems of equations to model and solve problems with two unknowns.

• Example: Population growth can be modeled by $P(t) = P_0 e^{rt}$ (exponential function).

Using Digital Technology in Mathematics

Students will utilize technology to evaluate expressions, graph functions, and solve equations and inequalities.

• Exact and Approximate Values: Use calculators or software to find decimal approximations of irrational numbers.

• Evaluating Formulas: Technology can be used to substitute values and compute results efficiently.

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

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

• Evaluating Logarithms: Use technology to compute logarithms, especially for non-integer values.

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

• Example: Use a calculator to approximate $\sqrt{2} \approx 1.414$.

Summary Table: Course Goals and SLOs

Goal	Student Learning Outcomes (SLOs)
Understand the concept of a function	Define relation and function Identify domain and range Recognize functions from representations Apply function notation Solve for intercepts and zeros Determine if a function is even or odd
Build new functions from existing functions	Perform graph transformations Combine functions (arithmetic, composition) Determine inverse functions
Build and analyze graphs	Relate equations, tables, and graphs Construct graphs using key features Analyze graphs for key features
Solve equations, inequalities, and systems	Solve equations algebraically and graphically Solve inequalities algebraically and graphically Solve systems of linear equations
Master algebraic techniques and modeling	Interpret functions in context Write equations of lines Solve applications with various functions Solve applications with function operations and systems
Use digital technology in mathematics	Evaluate expressions and approximations Evaluate formulas and functions Graph functions Solve equations and inequalities Evaluate logarithms Perform matrix operations