Comprehensive Study Guide: College Algebra Foundations and Key Topics

Month 1 – Foundations

Equations, Inequalities, and Modeling

This section introduces the fundamental concepts of linear equations, inequalities, and mathematical modeling, which are essential for understanding algebraic relationships and solving real-world problems.

• Linear Equations: Equations of the form $ax + b = 0$, where a and b are constants.

• Applications: Word problems involving rates, mixtures, and proportions.

• Absolute Value Equations/Inequalities: Equations involving $|x|$; solutions require considering both positive and negative cases.

Example: Solve $2x - 5 = 9$.

• Add 5 to both sides: $2x = 14$

• Divide by 2: $x = 7$

Graphs and Cartesian Plane

Understanding the Cartesian plane and graphing equations is crucial for visualizing algebraic relationships and interpreting solutions.

• Cartesian Plane: A two-dimensional plane defined by the x-axis (horizontal) and y-axis (vertical).

• Lines and Circles: Standard forms: Line: $y = mx + b$; Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Graphing Equations: Plotting points and interpreting intercepts and slopes.

Example: Graph $y = 2x + 1$ by plotting the y-intercept (0,1) and using the slope 2.

Functions and Graphs (Part I)

This section covers the definition and properties of functions, including function notation and transformations.

• Definition of Function: A relation where each input has exactly one output.

• Domain and Range: The set of possible inputs (domain) and outputs (range).

• Function Notation: $f(x)$ represents the output when input is $x$.

• Transformations: Shifts, reflections, stretches, and compressions of graphs.

Example: If $f(x) = x^2$, then $f(3) = 9$.

Month 2 – Functions and Polynomials

Functions and Graphs (Part II)

Explores advanced function concepts, including transformations, composition, and inverse functions.

• Transformations: $f(x) + c$ (vertical shift), $f(x + c)$ (horizontal shift), $-f(x)$ (reflection).

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

• Inverse Functions: $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$

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

Polynomial and Rational Functions (Part I & II)

Focuses on the properties, graphs, and operations of polynomial and rational functions.

• Polynomial Functions: Functions of the form $f(x) = a_nx^n + \ldots + a_1x + a_0$

• Zeros: Solutions to $f(x) = 0$

• Rational Functions: Ratios of polynomials, $f(x) = \frac{P(x)}{Q(x)}$

• Asymptotes: Lines the graph approaches but never touches (vertical, horizontal, oblique)

• Division: Long and synthetic division for polynomials

Example: Find the zeros of $f(x) = x^2 - 4$.

• Set $x^2 - 4 = 0$

• Factor: $(x - 2)(x + 2) = 0$

• So, $x = 2$ or $x = -2$

Month 3 – Advanced Topics and Review

Exponential and Logarithmic Functions (Part I & II)

Examines exponential growth/decay and the properties and applications of logarithms.

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

• Properties: $b^{x+y} = b^x b^y$, $b^{x-y} = \frac{b^x}{b^y}$

• Logarithmic Functions: $y = \log_b x$ is the inverse of $b^y = x$

• Solving Equations: Use logarithms to solve $a \cdot b^x = c$

Example: Solve $2^x = 8$.

• Rewrite as $2^x = 2^3$

• So, $x = 3$

Systems of Equations and Inequalities

Introduces methods for solving systems of linear and nonlinear equations and inequalities.

• Substitution and Elimination: Techniques for solving systems of equations.

• Matrices: Representing and solving systems using matrix operations.

• Nonlinear Systems: Systems involving quadratic or other non-linear equations.

Example: Solve the system: $x + y = 5$, $x - y = 1$.

• Add equations: $2x = 6 \Rightarrow x = 3$

• Substitute: $3 + y = 5 \Rightarrow y = 2$

Sequences, Series, and Probability

Covers arithmetic and geometric sequences, series notation, and basic probability concepts.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Series Notation: $\sum_{i=1}^n a_i$

• Binomial Theorem: $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

Example: Find the 5th term of the sequence $a_n = 2 + 3(n-1)$.

• $a_5 = 2 + 3(5-1) = 2 + 12 = 14$

Conic Sections & Analytic Geometry

Explores the properties and equations of circles, ellipses, hyperbolas, and parabolas.

• Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

• Parabola: $y = a(x - h)^2 + k$

Example: The equation $x^2 + y^2 = 9$ represents a circle with center (0,0) and radius 3.

Study Notes – Tips

• Active Notes: For each section, create summary cards with key formulas and 2 worked examples.

• Practice Problems: Complete at least 15–20 problems per section, including odd and challenge problems.

• Weekly Mini-Quizzes: Make 5–10 problems from the week’s topics for self-testing.

• Formula Sheet: Update weekly; keep a running sheet for quick review before the final.

• Tech Check: Practice with Desmos or a graphing calculator for visualization.

Summary Table: Key Algebraic Functions

Function Type	General Form	Key Features
Linear	$y = mx + b$	Slope $m$, y-intercept $b$
Quadratic	$y = ax^2 + bx + c$	Vertex, axis of symmetry, parabola shape
Polynomial	$y = a_nx^n + \ldots + a_0$	Degree $n$, zeros, end behavior
Rational	$y = \frac{P(x)}{Q(x)}$	Asymptotes, holes, domain restrictions
Exponential	$y = a b^x$	Growth/decay, horizontal asymptote
Logarithmic	$y = \log_b x$	Inverse of exponential, vertical asymptote