Flashcard 1 - Term: Pythagorean Theorem
Definition: Formula: a² + b² = c² (where c is the hypotenuse). Example: If a right triangle has legs of 3 and 4, the hypotenuse is √(3² + 4²) = √25 = 5. Used for: Finding distances, solving right triangle problems.
Explanation: The Pythagorean theorem relates the sides of a right triangle. It's one of the most fundamental formulas in geometry and appears in coordinate geometry, trigonometry, and physics.
Bloom Level: Remember
Flashcard 2 - Term: Slope Formula
Definition: Formula: m = (y₂ - y₁)/(x₂ - x₁). Example: Slope between (2,3) and (5,9) is (9-3)/(5-2) = 6/3 = 2. Used for: Finding rate of change, graphing lines, parallel/perpendicular lines.
Explanation: Slope measures how steep a line is and represents rate of change. Positive slope rises left to right; negative slope falls left to right. Zero slope is horizontal.
Bloom Level: Remember
Flashcard 3 - Term: Quadratic Formula
Definition: Formula: x = [-b ± √(b² - 4ac)] / 2a. Example: Solve x² + 5x + 6 = 0: x = [-5 ± √(25-24)]/2 = [-5 ± 1]/2 = -2 or -3. Used for: Solving any quadratic equation.
Explanation: The quadratic formula solves ax² + bx + c = 0 for x. The discriminant (b² - 4ac) tells you if there are two real solutions (positive), one (zero), or two complex solutions (negative).
Bloom Level: Remember
Flashcard 4 - Term: Area of a Circle
Definition: Formula: A = πr². Example: Circle with radius 4 has area π(4)² = 16π ≈ 50.27 square units. Used for: Geometry problems, real-world applications involving circular regions.
Explanation: Circle area grows with the square of the radius. Doubling the radius quadruples the area. Remember: circumference is 2πr (linear), area is πr² (quadratic).
Bloom Level: Remember
Flashcard 5 - Term: Derivative Power Rule
Definition: Formula: d/dx[xⁿ] = nxⁿ⁻¹. Example: d/dx[x³] = 3x². d/dx[x⁵] = 5x⁴. Used for: Finding rates of change, optimization, curve sketching.
Explanation: The power rule is the foundation of differentiation. Bring down the exponent as a coefficient, then subtract 1 from the exponent. Works for any real number n.
Bloom Level: Remember
Flashcard 6 - Term: Distance Formula
Definition: Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]. Example: Distance from (1,2) to (4,6) is √[(4-1)² + (6-2)²] = √[9+16] = 5. Derived from: Pythagorean theorem.
Explanation: Distance formula finds the straight-line distance between two points in the coordinate plane. It's the Pythagorean theorem applied to coordinate geometry.
Bloom Level: Remember
Flashcard 7 - Term: Midpoint Formula
Definition: Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2). Example: Midpoint of (2,3) and (8,7) is ((2+8)/2, (3+7)/2) = (5,5). Used for: Finding centers, bisecting segments.
Explanation: The midpoint formula averages the x-coordinates and y-coordinates separately. It finds the point exactly halfway between two given points.
Bloom Level: Remember
Flashcard 8 - Term: Standard Deviation
Definition: Concept: Measures spread of data from the mean. Formula: σ = √[Σ(x-μ)²/N]. Higher σ = more spread out data. Used in: Statistics, analyzing variability, quality control.
Explanation: Standard deviation quantifies how much data values deviate from the average. About 68% of data falls within 1 standard deviation of the mean in a normal distribution.
Bloom Level: Understand
Flashcard 9 - Term: Factoring Difference of Squares
Definition: Pattern: a² - b² = (a+b)(a-b). Example: x² - 9 = (x+3)(x-3). 25y² - 16 = (5y+4)(5y-4). Used for: Simplifying expressions, solving equations.
Explanation: Difference of squares is one of the most useful factoring patterns. Recognize it by: two perfect squares separated by subtraction.
Bloom Level: Remember
Flashcard 10 - Term: Properties of Exponents
Definition: Rules: x^a · x^b = x^(a+b), (x^a)^b = x^(ab), x^a / x^b = x^(a-b), x^0 = 1, x^(-a) = 1/x^a. Example: x³ · x⁴ = x⁷.
Explanation: Exponent rules simplify algebraic expressions. When multiplying like bases, add exponents. When raising a power to a power, multiply exponents.
Bloom Level: Remember
Flashcard 11 - Term: Sum of Interior Angles (Polygon)
Definition: Formula: (n-2) × 180° where n = number of sides. Example: Pentagon (5 sides): (5-2) × 180° = 540°. Hexagon: (6-2) × 180° = 720°.
Explanation: Any polygon can be divided into (n-2) triangles from one vertex. Since each triangle has 180°, multiply by the number of triangles.
Bloom Level: Remember
Flashcard 12 - Term: Volume of a Cylinder
Definition: Formula: V = πr²h (area of base × height). Example: Cylinder with radius 3 and height 5: V = π(3)²(5) = 45π ≈ 141.37 cubic units.
Explanation: Cylinder volume is base area times height. Since the base is a circle (πr²), multiply by height h. Works for any prism: base area × height.
Bloom Level: Remember
