College Algebra: Functions and Modeling – Structured Study Notes

Course Overview: Functions and Modeling (MATH 1483)

Course Description

This course introduces students to the general concepts of relations and functions, focusing on polynomial, exponential, and logarithmic functions. Students will learn to solve equations, analyze graphs, and apply algebraic techniques to real-world problems.

• Prerequisites: Completion of introductory algebra courses or placement by assessment.

• Course Competencies: Identify and graph functions, solve equations, analyze models, and apply algebraic methods to practical problems.

Grading Policy

• Grades are based on overall average according to the following scale:

Overall Average	Course Grade
90 – 100%	A
80 – 89%	B
70 – 79%	C
60 – 69%	D
Below 60%	F

Module One: Functions, Linear Behavior & Modeling, Zeros, Literal Equations

Key Concepts

This module covers the foundational ideas of functions, their representations, and applications in modeling linear relationships.

• Function: A relation that assigns each input exactly one output.

• Graphing Functions: Visual representation of functions using coordinate axes.

• Linear Functions: Functions of the form $f(x) = mx + b$.

• Intercepts: Points where the graph crosses the axes.

• Slope: Rate of change; calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Modeling: Using functions to represent real-world situations.

Example: The cost of a taxi ride can be modeled by a linear function: $C(x) = 2.50x + 3$, where $x$ is the number of miles.

Section Breakdown

• Functions and Models: Identifying functions from verbal descriptions and analyzing function graphs.

• Graphs of Functions: Creating scatterplots and interpreting data sets.

• Linear Functions: Finding intercepts, slope, and applying linear models to real-world problems.

• Equations of Lines: Writing equations in slope-intercept and point-slope forms.

• Algebraic and Graphical Solution of Linear Equations: Solving equations and interpreting solutions graphically.

Module Two: Quadratic, Piecewise, Absolute Value, and Other Functions

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written as $f(x) = ax^2 + bx + c$.

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Factoring: Expressing quadratics as products of binomials.

• Graphing: Parabolas open upward if $a > 0$, downward if $a < 0$.

• Discriminant: $D = b^2 - 4ac$ determines the nature of roots.

• Applications: Projectile motion, revenue/cost models.

Example: The height of a ball thrown upward can be modeled by $h(t) = -16t^2 + 32t + 5$.

Piecewise and Absolute Value Functions

• Piecewise Functions: Defined by different expressions over different intervals.

• Absolute Value Functions: $f(x) = |x|$; graph is V-shaped.

Example: $f(x) = \begin{cases} x+2 & x < 0 \\ 2x-1 & x \geq 0 \end{cases}$

Module Three: Transformations, Symmetry, Composition of Functions

Transformations of Functions

Transformations alter the graph of a function by shifting, stretching, compressing, or reflecting.

• Vertical Shifts: $f(x) + k$ shifts up/down.

• Horizontal Shifts: $f(x-h)$ shifts right/left.

• Reflections: $-f(x)$ reflects over x-axis; $f(-x)$ reflects over y-axis.

• Stretch/Compression: $af(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

Symmetry: Even functions satisfy $f(-x) = f(x)$; odd functions satisfy $f(-x) = -f(x)$.

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

Module Four: Inverse, Exponential, Logarithmic, and Logistic Functions

Exponential and Logarithmic Functions

Exponential functions model rapid growth or decay, while logarithmic functions are their inverses.

• Exponential Function: $f(x) = a \cdot b^x$

• Logarithmic Function: $f(x) = \log_b(x)$

• Inverse Functions: $f^{-1}(x)$ reverses the effect of $f(x)$.

• Properties of Logarithms:

  • $\log_b(xy) = \log_b(x) + \log_b(y)$

  • $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

  • $\log_b(x^r) = r \log_b(x)$

• Applications: Compound interest, population growth, radioactive decay.

Example: The amount in an account after $t$ years with principal $P$ and annual rate $r$ compounded $n$ times per year: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

Logistic Functions

• Logistic Function: $f(x) = \frac{L}{1 + e^{-k(x-x_0)}}$ models population growth with a carrying capacity.

Example: Modeling the spread of a virus in a population.

Course Activities and Resources

Learning Resources

• MyMathLab: Online homework, tutorials, and multimedia resources.

• Textbook: College Algebra in Context (optional).

• Math Lab: On-campus support for homework and exam preparation.

Exam Preparation Tips

• Review all notes and textbook pages.

• Re-work homework problems.

• Practice with module review problems.

• Seek help from instructors or math lab staff if needed.

Summary Table: Key Function Types

Function Type	General Form	Key Properties
Linear	$f(x) = mx + b$	Constant rate of change (slope)
Quadratic	$f(x) = ax^2 + bx + c$	Parabola, vertex, axis of symmetry
Exponential	$f(x) = a \cdot b^x$	Rapid growth/decay
Logarithmic	$f(x) = \log_b(x)$	Inverse of exponential
Absolute Value	$f(x) = |x|$	V-shaped graph
Piecewise	Defined by intervals	Multiple expressions
Logistic	$f(x) = \frac{L}{1 + e^{-k(x-x_0)}}$	Growth with carrying capacity

Additional info: These notes expand on the syllabus outline, providing definitions, formulas, and examples for each major topic. For more detailed practice, refer to assigned textbook sections and MyMathLab modules.