Untitled Quiz

Basic Algebra Concepts

• Rational Numbers: Represented as a/b where a and b are integers; b cannot be zero.
• Irrational Numbers: Numbers that cannot be expressed as a fraction; they do not terminate or repeat.
• Real Numbers: The complete set that includes both rational and irrational numbers.

Equations and Expressions

• Equation: A mathematical statement asserting that two expressions are equal; solved to find unknown values.
• Expression: A combination of symbols representing a quantity; simplified rather than solved.

Functions and Relations

• Function: A specific type of relation where each input corresponds to one unique output.
• Relation: A collection of inputs paired with outputs, often represented as ordered pairs.

Domain and Range

• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values from a function.

Graphical Analysis

• Vertical Line Test: A criterion to determine if a relation is a function; no vertical line should intersect a graph more than once.
• Function Notation: Expressed typically as f(x) = mx + b, where m represents the slope and b the y-intercept.

Types of Functions

• Parent Function: The simplest form of a function, with linear as y = x and quadratic as y = x².
• Discrete Function: Represented graphically as a scatter plot.
• Continuous Function: A function that includes an infinite number of points closely packed together.

Slope and Lines

• Slope Equation: Calculated as (y2 - y1) / (x2 - x1).
• Types of Slopes: Can be positive, negative, zero (horizontal), or undefined (vertical).
• Vertical Line: Represented by the equation x = b.
• Horizontal Line: Represented by the equation y = a.
• Parallel Lines: Lines with the same slope but different y-intercepts.
• Perpendicular Lines: Lines that intersect at a right angle; slopes are negative reciprocals.

Equation Forms

• Point-Slope Form: Expressed as y - y1 = m(x - x1), representing a line's steepness at a point.
• Direct Variation: In the form y = ax, where 'a' is a constant that indicates the variation rate.
• Constant Variation: The non-zero constant 'a' in the direct variation equation.

Linear Models and Predictions

• Direct Variation Line: Has slope 'a' with a y-intercept of zero.
• Linear Regression: A method for modeling the relationship between two variables with a linear equation.
• Extrapolation: Making predictions beyond the current dataset using established models.
• Interpolation: Using a model to estimate data values within the existing dataset.

Additional Concepts

• Model: A representation in equation form that describes a relationship.
• Correlation Coefficient (r): Indicates the strength and direction of a linear relationship; r = 0 indicates no correlation, ±1 indicates perfect correlation.
• Line of Least Squares: A line that minimizes the sum of the squared differences between observed values and the line.
• Slope in Line of Least Squares: Determined through a specific formula involving sums of products and squared terms.
• Piecewise Definition of Absolute Value: |x| = x if x ≥ 0; = -x if x < 0.
• Transformations of Absolute Value Functions:
  • y = |x| + k: Affects the y-intercept.
  • y = |x + k|: Affects the x-intercept.
• Greatest Integer Function: Denoted as [x], representing the largest integer less than or equal to x, also known as the floor function; its opposite is the ceiling function, which rounds up.