CLEP College Mathematics cheat sheet

Slope between two points: m = (y2 − y1)/(x2 − x1); horizontal line m = 0, vertical line m undefined.

Slope-intercept form y = mx + b; point-slope form y − y1 = m(x − x1); standard form Ax + By = C has slope −A/B.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m1·m2 = −1).

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac))/(2a). Discriminant b² − 4ac: positive = two real roots, zero = one repeated root, negative = no real roots.

Vertex of y = ax² + bx + c lies at x = −b/(2a); the parabola opens upward if a > 0 and downward if a < 0.

Function definition: each input in the domain maps to exactly one output. f(g(x)) means substitute g(x) into f; f⁻¹ undoes f, so f(f⁻¹(x)) = x.

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

Log rules: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n·log a; y = log_b x is equivalent to bʸ = x.

Exponential growth/decay: y = a·bˣ grows when b > 1 and decays when 0 < b < 1; compound-style models use y = a(1 + r)ˣ.

Multiplication principle: if a task has m ways for one step and n ways for a second, the sequence has m·n ways.

Permutations count ordered arrangements: nPr = n!/(n − r)!. Combinations count unordered selections: nCr = n!/(r!(n − r)!).

Use permutations when order matters (finishing places, seating in a row); use combinations when order does not (committees, card hands).

Probability of an event: P(E) = (favorable outcomes)/(total equally likely outcomes); 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E). For any two events, P(A or B) = P(A) + P(B) − P(A and B).

Mutually exclusive events have P(A and B) = 0, so P(A or B) = P(A) + P(B).

Independent events: P(A and B) = P(A)·P(B). Conditional probability: P(A|B) = P(A and B)/P(B).

With replacement the selections are independent; without replacement the denominator shrinks and outcomes are dependent.

Circular arrangements of n distinct objects number (n − 1)!; arrangements of a multiset use n!/(n1!·n2!·…).

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

Range = maximum − minimum. Standard deviation measures typical distance from the mean; adding a constant to every value leaves the standard deviation unchanged.

For a data set, roughly 68% of values lie within one standard deviation of the mean and about 95% within two when the distribution is bell-shaped.

z-score = (value − mean)/(standard deviation); it states how many standard deviations a value sits from the mean.

Quartiles split ordered data into four parts; interquartile range IQR = Q3 − Q1; a common outlier rule flags values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Correlation r ranges from −1 to 1; r near 1 or −1 indicates a strong linear association, r near 0 a weak one. Correlation does not prove causation.

A least-squares regression line has slope b = r·(sy/sx) and passes through the point (x̄, ȳ).

Sampling bias arises when the selection method favors some part of the population; a random sample reduces bias, while a larger sample mainly reduces variability.

Simple interest: I = Prt, where P is principal, r the annual rate as a decimal, and t the time in years. Total amount A = P(1 + rt).

Compound interest: A = P(1 + r/n)^(nt), where n is the number of compounding periods per year. More frequent compounding raises the effective yield.

Continuous compounding: A = P·e^(rt).

Effective annual rate: EAR = (1 + r/n)ⁿ − 1; it converts a nominal rate with n compoundings into a comparable yearly rate.

Percent change = (new − old)/old × 100%; a price raised by p% then lowered by p% ends below the starting value.

Markup and markdown: selling price = cost × (1 + markup rate); sale price = original × (1 − discount rate).

A percentage increase of p% multiplies by (1 + p/100); a decrease of p% multiplies by (1 − p/100).

Average daily balance or unit-price comparisons: divide total cost by total quantity to compare offers on a common basis.

Doubling time under compound growth is approximated by the rule of 72: years ≈ 72 ÷ (annual percentage rate).

Perimeter and area: rectangle A = lw; triangle A = ½bh; circle circumference C = 2πr and area A = πr².

Pythagorean theorem for right triangles: a² + b² = c², where c is the hypotenuse.

Volume: rectangular solid V = lwh; cylinder V = πr²h; cone V = ⅓πr²h; sphere V = (4/3)πr³.

Sum of interior angles of an n-sided polygon = (n − 2)·180°; each interior angle of a regular n-gon measures (n − 2)·180°/n.

Similar figures have proportional sides and equal angles; areas scale by the square of the scale factor and volumes by its cube.

Complementary angles sum to 90°, supplementary angles to 180°; vertical angles are equal.

In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

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

Slope of a line through two points equals rise over run; slope is undefined for a vertical line.

Union A ∪ B contains elements in A or B or both; intersection A ∩ B contains elements in both; complement A′ contains elements of the universal set not in A.

Inclusion–exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For three sets, subtract each pairwise intersection and add back the triple intersection.

Disjoint sets have A ∩ B = ∅, so n(A ∪ B) = n(A) + n(B).

Subset: A ⊆ B means every element of A is in B; the empty set is a subset of every set.

De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Conditional statement p → q is false only when p is true and q is false. Its contrapositive, ∼q → ∼p, is logically equivalent to it.

Converse q → p and inverse ∼p → ∼q are not equivalent to the original conditional.

A biconditional p ↔ q is true exactly when p and q share the same truth value.

Valid argument forms: modus ponens (p → q, p, therefore q) and modus tollens (p → q, ∼q, therefore ∼p).

Divisibility: by 3 if digit sum is divisible by 3; by 9 if digit sum is divisible by 9; by 4 if the last two digits form a multiple of 4; by 6 if divisible by both 2 and 3.

A prime has exactly two positive divisors, 1 and itself; 1 is not prime and 2 is the only even prime.

Prime factorization writes a number as a product of primes; the greatest common factor takes the lowest power of each shared prime, the least common multiple the highest power of each prime present.

Rational numbers can be written as a/b with integers a and b ≠ 0; irrational numbers (√2, π) cannot.

Order of operations: parentheses, exponents, multiplication and division left to right, addition and subtraction left to right.

Absolute value |x| gives distance from zero; |x| = k with k > 0 has solutions x = k and x = −k.

Scientific notation a × 10ⁿ requires 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Percent means per hundred: p% = p/100. To convert a fraction to a percent, divide then multiply by 100.