NVO Grade 10 Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For any equation ax² + bx + c = 0 with a not equal to zero, the solutions are x = (−b ± √(b² − 4ac)) divided by 2a. It solves every quadratic, including those that do not factor.

Discriminant

The expression b² − 4ac from a quadratic ax² + bx + c. Positive means two distinct real roots, zero means one repeated real root, negative means no real roots because the square root is undefined over the reals.

Vertex of a parabola

The turning point of y = ax² + bx + c, located at x = −b/(2a). Its y-value is found by substitution. It is a minimum when a > 0 and a maximum when a < 0.

Composite function

The function f(g(x)), formed by applying g first and then f to the result. Composition is not generally commutative, so f(g(x)) and g(f(x)) usually give different outputs and may have different domains.

Inverse function

A function f⁻¹ that undoes f, so f⁻¹(f(x)) = x. It exists only when f is one-to-one. Its graph is the reflection of f's graph across the line y = x.

Change of base formula

log_b x equals ln x divided by ln b, or equivalently log x / log b in any common base. It allows evaluation of logarithms whose base is not available on a calculator.

Exponential growth model

y = a·bˣ, where a is the initial amount and b is the growth factor per period. When b > 1 the quantity grows; the percent increase per period equals (b − 1) × 100%.

Logarithm product rule

The logarithm of a product equals the sum of the logarithms: log_b(xy) = log_b x + log_b y. Related rules handle quotients by subtraction and powers by multiplying the exponent outside.

Distance formula

The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²). It follows from the Pythagorean theorem applied to the horizontal and vertical differences.

Midpoint formula

The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2). It averages the x-coordinates and separately averages the y-coordinates of the endpoints.

Perpendicular slope relationship

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the slopes equals −1. A horizontal line and a vertical line are also perpendicular.

Circle equation

A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r². Expanding this form produces a quadratic equation in x and y.

Pythagorean theorem

In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse: a² + b² = c². Its converse identifies a right angle when this relationship holds.

Similar figure ratios

When two figures are similar with side-length ratio k, corresponding angles are equal, the ratio of areas is k², and the ratio of volumes is k³. These ratios follow from scaling each dimension.

Sector area

The area of a sector with central angle θ in degrees is (θ/360) × πr². It is the same fraction of the full circle's area as the angle is of 360 degrees.

Sphere volume

The volume enclosed by a sphere of radius r is (4/3)πr³. Its surface area is 4πr². Volume grows with the cube of the radius while surface area grows with the square.

Rational number

A number expressible as a fraction a/b where a and b are integers and b is not zero. Its decimal expansion either terminates or repeats. Integers and terminating decimals are rational.

Irrational number

A real number that cannot be written as a ratio of two integers. Its decimal expansion never terminates and never repeats. Familiar examples include √2, √3, and π.

Rational exponent rule

A fractional exponent means a radical: a^(m/n) equals the nth root of a raised to the m power. For even n, the base a must be nonnegative for the expression to be real.

Scientific notation

A number written as c × 10ⁿ, where c is at least 1 and less than 10 in absolute value. Multiplying such numbers adds the exponents; dividing subtracts the exponents of the powers of ten.

Percent change formula

Percent change equals (new value − old value) divided by old value, then multiplied by 100. A positive result is an increase; a negative result indicates a decrease relative to the original amount.

Absolute value equation

The equation |x − a| = d means x lies exactly d units from a on the number line, giving x = a + d or x = a − d. It has two solutions when d > 0 and one when d = 0.

Dimensional analysis

A conversion method that multiplies a quantity by ratios equal to one, arranged so unwanted units cancel. For example, multiplying by 1000 m / 1 km converts kilometers to meters.

Standard deviation

A measure of how far data values typically lie from their mean. Adding a constant to every value leaves it unchanged; multiplying every value by k multiplies it by the absolute value of k.

Interquartile range

The difference between the third quartile and the first quartile, IQR = Q3 − Q1. It measures the spread of the middle half of the data and is resistant to extreme values.

Outlier rule

A data value is a potential outlier if it lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR. Outliers strongly affect the mean and standard deviation but barely affect the median.

Complement rule

For any event E, the probability that E does not occur is 1 − P(E). The event and its complement together cover every outcome, so their probabilities sum to exactly one.

Addition rule

P(A or B) = P(A) + P(B) − P(A and B). The overlap is subtracted so outcomes in both events are not counted twice. For mutually exclusive events the overlap is zero.

Conditional probability

The probability of B given that A occurred is P(B|A) = P(A and B) / P(A), provided P(A) > 0. It restricts the sample space to outcomes consistent with event A.

Expected value

For a discrete random variable, the expected value is the sum of each outcome multiplied by its probability. It represents the long-run average result over many repetitions of the experiment.

Linear model interpretation

In y = mx + b, the slope m is the constant change in y per one-unit increase in x, and the intercept b is the predicted y when x equals zero, when that value is meaningful.

Choosing a model from data

Constant first differences between consecutive outputs suggest a linear model; constant second differences suggest a quadratic model; constant ratios of consecutive outputs suggest an exponential model.