EN VIII Mathematics flashcards
EN VIII Mathematics · Build recall with a focused review session.
32 cards. Read the term and the definition here. The deck at the top of this page is optional.
- Order of operations
- Evaluate grouped expressions first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.
- Divisibility by 3
- An integer is divisible by 3 exactly when the sum of its digits is divisible by 3; the same digit-sum test works for divisibility by 9.
- Reciprocal division rule
- Dividing by a fraction equals multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c, valid whenever c and d are nonzero.
- Percent change formula
- Percent change equals (new value − old value) divided by the old value, times 100; a positive result is an increase, a negative result a decrease.
- Negative exponent rule
- For a nonzero base a, a raised to the power −n equals 1 divided by a raised to the power n, so a^(−n) = 1/a^n.
- Absolute value
- The absolute value of a number is its distance from zero on the number line, so it is always nonnegative: |x| = x if x ≥ 0 and |x| = −x if x < 0.
- Slope formula
- The slope through two points (x₁,y₁) and (x₂,y₂) is the change in y divided by the change in x: m = (y₂ − y₁)/(x₂ − x₁).
- Quadratic formula
- The solutions of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)]/(2a), obtained by completing the square on the general quadratic.
- Discriminant
- For ax² + bx + c = 0, the discriminant b² − 4ac determines the roots: positive gives two real roots, zero gives one repeated real root, negative gives no real roots.
- Parabola vertex
- The graph of y = ax² + bx + c is a parabola whose vertex has x-coordinate −b/(2a); it opens upward if a > 0 and downward if a < 0.
- Perpendicular slopes
- Two nonvertical lines are perpendicular exactly when the product of their slopes is −1, meaning each slope is the negative reciprocal of the other.
- Composite function
- The composite f(g(x)) is formed by applying g first and then f to the result; its domain requires x to be in g's domain and g(x) in f's domain.
- Exponential growth form
- The function y = a·b^x has initial value a at x = 0; it models growth when b > 1 and decay when 0 < b < 1.
- Triangle area formula
- The area of a triangle equals one half the base times the corresponding height: A = ½bh, where the height is perpendicular to the chosen base.
- Pythagorean theorem
- In a right triangle with legs a and b and hypotenuse c, a² + b² = c²; the converse states that this equation holding makes the triangle right.
- Polygon angle sum
- The interior angles of a polygon with n sides sum to (n − 2) × 180 degrees; for a regular polygon each interior angle is that sum divided by n.
- Circle area and circumference
- For a circle of radius r, the area is πr² and the circumference is 2πr; the diameter equals 2r and π is approximately 3.14159.
- Similar figure ratios
- Similar figures have equal corresponding angles and proportional sides; if the side ratio is k, areas scale by k² and volumes by k³.
- Distance formula
- The distance between points (x₁,y₁) and (x₂,y₂) in the coordinate plane is √[(x₂ − x₁)² + (y₂ − y₁)²], derived from the Pythagorean theorem.
- Inscribed angle theorem
- An angle inscribed in a circle measures half the intercepted arc, so an angle subtending a diameter is a right angle of 90 degrees.
- Cylinder volume
- The volume of a cylinder equals the base area times height: V = πr²h, where r is the base radius and h the perpendicular height.
- Mean, median, mode
- The mean is the sum divided by the count, the median is the middle of ordered data, and the mode is the most frequently occurring value.
- Interquartile range
- The interquartile range equals Q₃ − Q₁, the spread of the middle half of the data; it resists distortion by extreme values better than the range.
- Standard deviation meaning
- Standard deviation measures the typical distance of data values from their mean; a larger value indicates greater spread, and it is never negative.
- Complement rule
- For any event A, the probability that A does not occur is 1 minus the probability that A occurs: P(not A) = 1 − P(A).
- Independent events
- Events A and B are independent when the occurrence of one does not change the other's probability, so P(A and B) = P(A) × P(B).
- Mutually exclusive events
- Two events are mutually exclusive when they cannot both occur, so P(A or B) = P(A) + P(B) and P(A and B) = 0.
- Conditional probability
- The probability of A given that B occurred is P(A|B) = P(A and B)/P(B), defined whenever P(B) is greater than zero.
- Normal distribution rule
- In a normal distribution about 68% of values fall within one standard deviation of the mean and about 95% within two standard deviations.
- Correlation coefficient
- The correlation coefficient r ranges from −1 to 1 and measures the strength and direction of a linear relationship; a value near zero indicates little linear association.
- Random sample
- A sample is random when every member of the population has an equal chance of being selected, which supports generalizing sample results to the population.
- Outlier rule
- A data value is commonly treated as an outlier if it lies below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR, where IQR = Q₃ − Q₁.