Algebra 2 Final Exam Flashcards

Key Definitions in Algebra

• Relation: A collection of ordered pairs that defines a relationship between elements.
• Domain: Represents all possible x-coordinates from a relation.
• Range: Encompasses all possible y-coordinates from a relation.
• Function: A special type of relation where each domain element maps to exactly one range element.

Algebraic Properties

• FOIL: A method used to find the product of two binomials.
• Distributive Property: States that a(b + c) equals ab + ac, demonstrating distribution across addition.
• Commutative Property:
  • Addition: The sum remains unchanged irrespective of the order of addends.
  • Multiplication: The product remains unchanged irrespective of the order of factors.
• Associative Property:
  • Addition: Grouping of addends does not affect the sum.
  • Multiplication: Grouping of factors does not affect the product.
• Identity Properties:
  • Addition: Adding zero to any number leaves it unchanged.
  • Multiplication: Multiplying any number by one leaves it unchanged.
• Inverses:
  • Additive Inverse: The sum of a number and its negative equals zero.
  • Multiplicative Inverse: The product of a number and its reciprocal equals one.

Numbers and Sets

• Rational Numbers: Can be expressed as a fraction a/b where both a and b are integers.
• Irrational Numbers: Numbers whose decimal representation is non-terminating and non-repeating.
• Perfect Squares: Square roots that result in rational numbers.
• Integers: The set of whole numbers including negatives, zero, and positives.
• Whole Numbers: Non-negative integers starting from zero.
• Natural Numbers: Positive integers starting from one.

Algebraic Expressions and Equations

• Algebraic Expression: Combines numbers, variables, and operations, containing at least one variable.
• Linear Equation: Expresses a straight line graphically; typically takes the form Ax + By = C.
• Standard Form: Refers to the form Ax + By = C for linear equations.

Graphing Concepts

• X-Intercept: The point where the graph intersects the x-axis.
• Y-Intercept: The point where the graph intersects the y-axis.
• Slope: Indicates the steepness of a line, calculated as the ratio of vertical change to horizontal change, expressed by m=(y2-y1)/(x2-x1).

Types of Functions

• Constant Function: A function where output remains constant f(x)=c.
• Identity Function: Defined as y=x, reflecting that every input produces the same output.
• Absolute Value Function: Expressed as f(x)=|x|, which returns the distance from zero.
• Quadratic Function: Formulated as y=ax^2 + bx + c, resulting in a parabola on a graph.

Geometry and Data Analysis

• Parabola: The U-shaped graph produced by a quadratic equation.
• Axis of Symmetry: A vertical line through the vertex of the parabola, dividing it into two symmetrical halves.
• Vertex: The highest or lowest point on a parabola, located at the axis of symmetry.

Systems and Matrices

• System of Linear Equations: Consists of two or more linear equations involving the same variables.
• Matrix: A rectangular arrangement of numbers or variables presented in rows and columns, typically enclosed by brackets.

Graphing Relationships

• Positive Slope: Indicates an upward trend in a graph.
• Negative Slope: Indicates a downward trend in a graph.
• Zero Slope: Represents a horizontal line, indicating no change.
• Undefined Slope: Generally occurs with vertical lines where a change in x does not exist.

These notes consolidate essential algebra concepts and serve as a quick reference for study and revision.