NVO Grade 10 Mathematics cheat sheet

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a).

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 gives one repeated real root, D < 0 gives no real roots (two complex conjugates).

Vertex of y = ax² + bx + c lies at x = −b/(2a); the vertex is a maximum when a < 0 and a minimum when a > 0.

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, so f(g(x)) ≠ g(f(x)) in general.

Inverse of a function exists only if the function is one-to-one; to find f⁻¹, swap x and y then solve for y, and graph f⁻¹ as a reflection of f across y = x.

Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

Logarithms: log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, log_b(xⁿ) = n·log_b x; log_b x = ln x / ln b (change of base).

Exponential growth/decay model: y = a·bˣ, where a is the initial value, b > 1 gives growth, 0 < b < 1 gives decay, and the percent rate per period is |b − 1|·100%.

Systems of two linear equations have one solution when slopes differ, none when slopes are equal but intercepts differ, and infinitely many when the equations are equivalent.

Distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Circle: area A = πr², circumference C = 2πr; equation with center (h, k) and radius r is (x − h)² + (y − k)² = r².

Triangle area = ½bh; a triangle's interior angles sum to 180°, and an exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem: in a right triangle, a² + b² = c² where c is the hypotenuse; converse holds, so a² + b² = c² implies a right angle opposite c.

Volume formulas: cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³, rectangular prism V = lwh.

Similar figures: corresponding angles are equal and corresponding side lengths share a constant ratio k; area ratio is k² and volume ratio is k³.

Surface area of a sphere is 4πr²; lateral surface area of a cylinder is 2πrh, and total surface area adds the two circular bases.

Arc length = (θ/360°)·2πr and sector area = (θ/360°)·πr², where θ is the central angle in degrees.

Real numbers split into rational numbers (expressible as a/b with integers a, b and b ≠ 0) and irrational numbers (non-terminating, non-repeating decimals such as √2 and π).

A number is rational exactly when its decimal expansion terminates or repeats; irrational expansions never terminate and never repeat.

Radical simplification: √(ab) = √a·√b and √(a/b) = √a/√b for a, b ≥ 0; rationalize denominators by multiplying by the conjugate or the radical itself.

Rational exponents: a^(m/n) = ⁿ√(aᵐ), valid for a ≥ 0 when n is even; negative exponents give reciprocals, a^(−m/n) = 1/a^(m/n).

Scientific notation writes a number as c × 10ⁿ with 1 ≤ |c| < 10; multiplying adds exponents, dividing subtracts them.

Percent change = (new − old)/old × 100%; successive percent changes multiply factors rather than add percentages.

Absolute value |x| is the distance from x to 0 on the number line; |x − a| = d means x is exactly d units from a, giving x = a ± d.

Unit conversion uses dimensional analysis: multiply by fractions equal to 1 (e.g., 1000 m / 1 km) so unwanted units cancel.

Mean = sum of values ÷ count; median is the middle value of ordered data (average of the two middle values when n is even); mode is the most frequent value.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves standard deviation unchanged, while multiplying by k multiplies it by |k|.

Outliers are values far from the bulk of the data (commonly beyond 1.5 × IQR from the quartiles); the median and IQR resist outliers, while the mean and standard deviation do not.

Interquartile range IQR = Q3 − Q1; a value is a potential outlier if it is below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Probability of an event P(E) satisfies 0 ≤ P(E) ≤ 1; complement rule P(not E) = 1 − P(E).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the last term is zero.

Multiplication rule: P(A and B) = P(A)·P(B|A); for independent events P(A and B) = P(A)·P(B).

Conditional probability P(B|A) = P(A and B)/P(A), valid when P(A) > 0.

Expected value of a discrete variable is Σ x·P(x); the mean of a data set is the balance point, so deviations above and below it sum to zero.

Linear model y = mx + b: m is the constant rate of change per unit and b is the starting value when the input is zero.

Exponential model y = a·bˣ fits data with a constant percent change per unit; linear models fit data with a constant absolute change per unit.

Choosing a model: constant first differences indicate a linear pattern, constant second differences indicate a quadratic pattern, and constant ratios indicate an exponential pattern.

Rate problems use the relation distance = rate × time; work problems combine rates as reciprocals, 1/t_total = 1/t₁ + 1/t₂.

Function models must respect domain constraints from context, such as nonnegative time, whole-number counts, or a maximum capacity.

Interpolation estimates between known data points; extrapolation predicts outside the data range and is less reliable because the pattern may not continue.

Inverse variation y = k/x has a constant product xy = k; direct variation y = kx has a constant ratio y/x = k.

When solving a modeled equation, check the answer against the units and the real-world constraints, and report the result with appropriate rounding and units.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a).

Discriminant D = b² − 4ac: D > 0 gives two distinct real roots, D = 0 gives one repeated real root, D < 0 gives no real roots (two complex conjugates).

Vertex of y = ax² + bx + c lies at x = −b/(2a); the vertex is a maximum when a < 0 and a minimum when a > 0.

Function notation: f(g(x)) means substitute g(x) into f; composition is generally not commutative, so f(g(x)) ≠ g(f(x)) in general.

Inverse of a function exists only if the function is one-to-one; to find f⁻¹, swap x and y then solve for y, and graph f⁻¹ as a reflection of f across y = x.

Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ.

Logarithms: log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, log_b(xⁿ) = n·log_b x; log_b x = ln x / ln b (change of base).

Exponential growth/decay model: y = a·bˣ, where a is the initial value, b > 1 gives growth, 0 < b < 1 gives decay, and the percent rate per period is |b − 1|·100%.

Systems of two linear equations have one solution when slopes differ, none when slopes are equal but intercepts differ, and infinitely many when the equations are equivalent.

Distance between points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes, perpendicular lines have slopes whose product is −1.

Circle: area A = πr², circumference C = 2πr; equation with center (h, k) and radius r is (x − h)² + (y − k)² = r².

Triangle area = ½bh; a triangle's interior angles sum to 180°, and an exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem: in a right triangle, a² + b² = c² where c is the hypotenuse; converse holds, so a² + b² = c² implies a right angle opposite c.

Volume formulas: cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³, rectangular prism V = lwh.

Similar figures: corresponding angles are equal and corresponding side lengths share a constant ratio k; area ratio is k² and volume ratio is k³.

Surface area of a sphere is 4πr²; lateral surface area of a cylinder is 2πrh, and total surface area adds the two circular bases.

Arc length = (θ/360°)·2πr and sector area = (θ/360°)·πr², where θ is the central angle in degrees.

Real numbers split into rational numbers (expressible as a/b with integers a, b and b ≠ 0) and irrational numbers (non-terminating, non-repeating decimals such as √2 and π).

A number is rational exactly when its decimal expansion terminates or repeats; irrational expansions never terminate and never repeat.

Radical simplification: √(ab) = √a·√b and √(a/b) = √a/√b for a, b ≥ 0; rationalize denominators by multiplying by the conjugate or the radical itself.

Rational exponents: a^(m/n) = ⁿ√(aᵐ), valid for a ≥ 0 when n is even; negative exponents give reciprocals, a^(−m/n) = 1/a^(m/n).

Scientific notation writes a number as c × 10ⁿ with 1 ≤ |c| < 10; multiplying adds exponents, dividing subtracts them.

Percent change = (new − old)/old × 100%; successive percent changes multiply factors rather than add percentages.

Absolute value |x| is the distance from x to 0 on the number line; |x − a| = d means x is exactly d units from a, giving x = a ± d.

Unit conversion uses dimensional analysis: multiply by fractions equal to 1 (e.g., 1000 m / 1 km) so unwanted units cancel.

Mean = sum of values ÷ count; median is the middle value of ordered data (average of the two middle values when n is even); mode is the most frequent value.

Standard deviation measures typical distance from the mean; adding a constant to every value leaves standard deviation unchanged, while multiplying by k multiplies it by |k|.

Outliers are values far from the bulk of the data (commonly beyond 1.5 × IQR from the quartiles); the median and IQR resist outliers, while the mean and standard deviation do not.

Interquartile range IQR = Q3 − Q1; a value is a potential outlier if it is below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Probability of an event P(E) satisfies 0 ≤ P(E) ≤ 1; complement rule P(not E) = 1 − P(E).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the last term is zero.

Multiplication rule: P(A and B) = P(A)·P(B|A); for independent events P(A and B) = P(A)·P(B).

Conditional probability P(B|A) = P(A and B)/P(A), valid when P(A) > 0.

Expected value of a discrete variable is Σ x·P(x); the mean of a data set is the balance point, so deviations above and below it sum to zero.

Linear model y = mx + b: m is the constant rate of change per unit and b is the starting value when the input is zero.

Exponential model y = a·bˣ fits data with a constant percent change per unit; linear models fit data with a constant absolute change per unit.

Choosing a model: constant first differences indicate a linear pattern, constant second differences indicate a quadratic pattern, and constant ratios indicate an exponential pattern.

Rate problems use the relation distance = rate × time; work problems combine rates as reciprocals, 1/t_total = 1/t₁ + 1/t₂.

Function models must respect domain constraints from context, such as nonnegative time, whole-number counts, or a maximum capacity.

Interpolation estimates between known data points; extrapolation predicts outside the data range and is less reliable because the pattern may not continue.

Inverse variation y = k/x has a constant product xy = k; direct variation y = kx has a constant ratio y/x = k.

When solving a modeled equation, check the answer against the units and the real-world constraints, and report the result with appropriate rounding and units.