OpenStax College Algebra Ch 2 Flashcards

Coordinate Plane Quadrants

• Quadrant I: Both x and y coordinates are positive (+,+).
• Quadrant II: x-coordinate is negative and y-coordinate is positive (-,+).
• Quadrant III: Both x and y coordinates are negative (-,-).
• Quadrant IV: x-coordinate is positive and y-coordinate is negative (+,-).

Intercepts

• x-intercept: The x-coordinate where a graph crosses the x-axis; y-coordinate at this point is zero.
• y-intercept: The y-coordinate where a graph crosses the y-axis; x-coordinate at this point is zero.

Essential Formulas

• Distance Formula: Calculate distance between two points (x₁, y₁) and (x₂, y₂) using d = √[(x₂ - x₁)² + (y₂ - y₁)²].
• Midpoint Formula: Find the midpoint between two points by using ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-Slope Form: (y - y₁) = m(x - x₁), with m as the slope and (x₁, y₁) as a point on the line.
• Standard Form of a Line: Ax + By = C, with A and B as integers.

Line Properties

• Parallel Lines: Maintain the same slope and do not intersect.
• Perpendicular Lines: Intersect at right angles; slopes are negative reciprocals of each other.

Quadratics

• Quadratic Formula: x = [-b ± √(b² - 4ac)]/(2a) for solving quadratic equations.
• Square Root Property: For any real number n, if x² = n, then x = ±√n.
• Quadratic Equation: An equation of the form ax² + bx + c = 0, where a ≠ 0.
• Quadratic Form: Represented as au² + bu + c, where u is any expression in x.

Types of Equations

• Linear Equation: A straight line equation, typically in the form ax + b = 0.
• Rational Equation: Contains one or more rational expressions (fractions).
• Radical Equation: Involves variables under a radical symbol.
• Absolute Value Equation: Contains an absolute value expression.
• Linear Inequality: Formed by replacing the equal sign in a linear equation with an inequality symbol.
• Absolute Value Inequality: An inequality that includes an absolute value expression.

Solutions and Equations

• Extraneous Solutions: Solutions that are invalid in the context of the original equation.
• Identity Equation: True for all values of the variable.
• Conditional Equation: True for some values of the variable.
• Inconsistent Equation: An equation that leads to a false statement.

Coordinate System Basics

• Origin: The point (0,0), where the x-axis and y-axis intersect.

Polynomial Characteristics

• Polynomial Equation: Contains only polynomial terms.

Slope Concepts

• Slope: Defined as the change in y-values over the change in x-values.
• Zero Slope: Indicates a horizontal line.
• Undefined Slope: Indicates a vertical line.
• Horizontal Line Equation: Represented as y = b.
• Vertical Line Equation: Represented as x = a.