College Algebra (MAC 1105) - Comprehensive Study Guide

Course Overview

This study guide summarizes the key topics and competencies for MAC 1105 - College Algebra, as outlined in the course syllabus. The course covers foundational concepts in algebra and precalculus, focusing on equations, functions, graphs, and their applications. Emphasis is placed on problem-solving, critical thinking, and computational proficiency.

Course Competencies and Main Topics

Equations and Inequalities

Equations and inequalities form the basis of algebraic problem-solving. Mastery of these concepts is essential for understanding more advanced topics in mathematics.

• Absolute Value Equations and Inequalities: Techniques for solving equations and inequalities involving absolute values.

• Quadratic Equations: Methods include factoring, completing the square, and the quadratic formula.

• Radical Equations: Solving equations involving roots and exponents.

• Equations Quadratic in Form: Recognizing and solving equations that can be rewritten as quadratics.

Key Formulas:

• Quadratic Formula: $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Absolute Value Inequality (example): $|x - a| < b \implies a - b < x < a + b$

Example: Solve $|2x - 3| = 7$.

• Solution: $2x - 3 = 7$ or $2x - 3 = -7$; $x = 5$ or $x = -2$.

Complex Numbers

Complex numbers extend the real number system and are essential for solving equations with negative radicands.

• Definition: $i = \sqrt{-1}$, so $i^2 = -1$.

• Operations: Addition, subtraction, multiplication, and division of complex numbers.

• Simplifying Powers of $i$: $i^n$ cycles every 4 powers.

Example: Simplify $(2 + 3i) + (4 - 5i) = 6 - 2i$.

Functions and Their Properties

Functions are mathematical relationships that assign each input exactly one output. Understanding their properties is crucial for modeling and analysis.

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

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

• Functional Notation: $f(x)$ represents the output for input $x$.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Operations on Functions: Addition, subtraction, multiplication, division, and composition.

• Inverse Functions: Functions that "undo" each other.

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

• Symmetry: Even, odd, and neither.

Key Formulas:

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

• Inverse: $f^{-1}(f(x)) = x$ (when $f$ is one-to-one)

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

Graphing and Transformations

Graphing functions and understanding transformations are essential for visualizing mathematical relationships.

• Basic Graphs: Linear, quadratic, absolute value, radical, and power functions.

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

• Graphing Inverse Functions: Reflect across the line $y = x$.

• Piecewise Functions: Graph each piece over its specified interval.

Example: The graph of $f(x) = |x|$ is a "V" shape with vertex at the origin.

Quadratic Functions and Applications

Quadratic functions model many real-world phenomena and have distinctive parabolic graphs.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Axis of Symmetry: $x = -\frac{b}{2a}$

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

• Applications: Maximum/minimum problems, projectile motion, area optimization.

Example: For $f(x) = 2x^2 - 4x + 1$, vertex at $x = 1$, $f(1) = -1$.

Systems of Equations and Inequalities

Systems involve solving for multiple variables using multiple equations or inequalities.

• Methods: Substitution, elimination (addition), and graphical methods.

• Systems in Three Variables: Extension of methods to three equations.

• Systems of Inequalities: Solution is the region where all inequalities overlap.

• Applications: Mixture problems, optimization, intersection points.

Example: Solve: $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ Solution: $x = 3$, $y = 2$.

Polynomial and Rational Functions

Polynomial and rational functions generalize linear and quadratic functions and introduce new behaviors such as asymptotes and end behavior.

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

• Graphing: Identify zeros, end behavior, and turning points.

• Rational Functions: Ratios of polynomials; may have vertical, horizontal, or oblique asymptotes.

• Domain: All real numbers except where the denominator is zero.

• Solving Inequalities: Use sign charts and test intervals.

Example: $f(x) = \frac{1}{x-2}$ has a vertical asymptote at $x=2$.

Exponential and Logarithmic Functions

Exponential and logarithmic functions model growth, decay, and many natural phenomena.

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

• Logarithmic Functions: $f(x) = \log_b(x)$, the inverse of exponential functions.

• Properties of Logarithms: Product, quotient, and power rules.

• Solving Equations: Use logarithms to solve for exponents and vice versa.

• Applications: Exponential growth and decay, compound interest, population models.

Key Formulas:

• Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

• Power Rule: $\log_b(M^k) = k \log_b(M)$

• Exponential Growth/Decay: $N(t) = N_0 e^{kt}$

Example: Solve $2^x = 8$; $x = 3$.

Equations in Two Variables: Circles, Distance, and Midpoint

Equations in two variables can represent geometric objects such as lines and circles.

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

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Graphing Circles: Center at $(h, k)$, radius $r$.

Example: Find the equation of a circle with center $(2, -1)$ and radius $3$:$(x - 2)^2 + (y + 1)^2 = 9$.

Grading and Assessment

Assessment	Weight
Tests (5 total)	40%
Quizzes (5 total)	15%
Homework	25%
Final Exam	20%

Grade Scale

Letter Grade	Percentage
A	90 - 100%
B	80 - 89%
C	70 - 79%
D	60 - 69%
F	0 - 59%

Course Outline (by Section)

• 1.2 Quadratic Equations

• 1.3 Complex Numbers; Quadratic Equations in Complex Number System

• 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations

• 1.6 Equations and Inequalities Involving Absolute Value

• 2.1 The Distance and Midpoint Formulas

• 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry

• 2.4 Circles

• 3.1 Functions

• 3.2 The Graph of a Function

• 3.3 Properties of Functions

• 3.4 Library of Functions; Piecewise-defined Functions

• 3.5 Graphing Techniques; Transformations

• 4.3 Quadratic Functions and Their Properties

• 4.5 Inequalities Involving Quadratic Functions

• 5.1 Polynomial Functions

• 5.2 Graphing Polynomial Functions

• 5.3 Properties of Rational Functions

• 5.4 The Graph of a Rational Function

• 5.5 Polynomial and Rational Inequalities

• 6.1 Composite Functions

• 6.2 One-to-One Functions; Inverse Functions

• 6.3 Exponential Functions

• 6.4 Logarithmic Functions

• 6.5 Properties of Logarithms

• 6.6 Logarithmic and Exponential Equations

• 6.8 Exponential Growth and Decay Models

• 12.1 Systems of Linear Equations

Additional Information

• Calculator: A regular scientific calculator (TI-30X IIS, TI-30XIIB, or TI-30XA) is required.

• Attendance: Regular attendance is expected and correlates with academic success.

• Support Services: Tutoring, advisement, and disability services are available.

Additional info: This guide is based on the course syllabus and outlines the main topics and skills required for success in College Algebra (MAC 1105). For detailed examples and practice problems, refer to the course textbook and assigned homework.