Equations and Inequalities

Solving Equations Over the Complex Numbers

Equations involving polynomials, rational expressions, and logarithms are fundamental in algebra. Solutions may be real or complex, depending on the equation.

• Quadratic Equations: Use the quadratic formula to solve .

• Rational Equations: Clear denominators and solve for the variable.

• Logarithmic Equations: Use properties of logarithms to combine and solve for the variable.

• Exponential Equations: Take logarithms of both sides to solve for the variable.

• Example: Solve by rearranging to and applying the quadratic formula.

Solving Inequalities

Inequalities require finding the set of values that satisfy the given condition. Solutions are often expressed in interval notation.

• Linear Inequalities: Solve as you would equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• Absolute Value Inequalities: Split into two cases and solve each.

• Example: Solve by considering and .

Functions and Graphs

Basic Functions and Transformations

Understanding the domain, range, and transformations of functions is essential for graphing and analysis.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Transformations: Shifts, stretches, compressions, and reflections of the parent function.

• Example: For , the graph is shifted right by 3 and up by 2.

Intercepts and Asymptotes

Intercepts are where the graph crosses the axes. Asymptotes are lines the graph approaches but never touches.

• x-intercept: Set and solve for .

• y-intercept: Evaluate .

• Horizontal Asymptote: For rational functions, compare degrees of numerator and denominator.

• Example: For , the horizontal asymptote is .

Polynomial and Rational Functions

Vertex, Intercepts, and Axis of Symmetry

Quadratic functions are graphed as parabolas. The vertex is the maximum or minimum point.

• Vertex Formula: for .

• Axis of Symmetry: .

• Example: For , vertex at .

Factoring and Roots

Factoring polynomials helps find roots and simplify expressions.

• Factoring Quadratics: where and are roots.

• Example: .

Exponential and Logarithmic Functions

Properties and Equations

Exponential and logarithmic functions model growth and decay, and are inverses of each other.

• Exponential Form:

• Logarithmic Form:

• Change of Base Formula:

• Example: because .

Compound Interest

Compound interest is calculated using exponential functions.

• Formula:

• Where: = final amount, = principal, = annual rate, = number of times compounded per year, = years.

• Example: , , , .

Systems of Equations and Inequalities

Solving Systems

Systems of equations can be solved by substitution, elimination, or graphing.

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

• Graphing: Plot both equations and find intersection point(s).

• Example: Solve and .

Graphing Inequalities

Graph the solution set and express in interval notation.

• Example: is solved by isolating and graphing the solution.

Matrices and Determinants

Matrix Operations

Matrices are used to organize and solve systems of equations.

• Addition: Add corresponding elements.

• Multiplication: Multiply rows by columns.

• Determinant: For a matrix , .

• Example: , .

Conic Sections

Circle and Parabola Equations

Conic sections include circles, ellipses, parabolas, and hyperbolas.

• Circle:

• Parabola:

• Vertex:

• Example: Find the center and radius from .

Sequences, Induction, and Probability

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers following a specific pattern.

• Arithmetic Sequence:

• Geometric Sequence:

• Example: Find the 5th term of .

Probability

Probability measures the likelihood of an event.

• Formula:

• Example: Probability of rolling a 3 on a six-sided die is .

Additional Topics

Difference Quotient

The difference quotient is used to find the average rate of change of a function.

• Formula:

• Example: For , .

Logarithmic Properties

Logarithms have several properties useful for simplifying expressions.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Example: because .

Summary Table: Key Algebraic Concepts

Graphing and Analysis

Graphing Functions

Graphing involves plotting points, identifying intercepts, asymptotes, and analyzing behavior.

• Increasing/Decreasing: Where the function rises or falls as increases.

• Symmetry: Even functions are symmetric about the y-axis; odd functions about the origin.

• Example: is even; is odd.

Application Problems

Word Problems

Algebra is used to model and solve real-world problems, such as geometry, finance, and motion.

• Example: The height of a ball thrown upward can be modeled by .

• Geometry: Perimeter and area problems often require setting up and solving equations.

Additional info: These notes expand upon the exam review questions, providing definitions, formulas, and examples for each major College Algebra topic represented in the file.