FTCE Mathematics 6-12 cheat sheet

Divisibility: by 3 iff digit sum is a multiple of 3; by 4 iff last two digits form a multiple of 4; by 9 iff digit sum is a multiple of 9; by 11 iff the alternating digit sum is a multiple of 11.

Rational numbers have terminating decimal expansions exactly when the reduced denominator has no prime factors other than 2 and 5.

For real a, b ≥ 0: √(ab) = √a·√b, and √(a/b) = √a/√b; rationalize denominators by multiplying by the conjugate.

Rules of exponents: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(m/n) = ⁿ√(a^m) for a > 0.

Absolute value: |xy| = |x||y|, |x| ≥ 0, and |x| = k (k > 0) means x = k or x = −k.

Complex numbers: i² = −1; (a + bi)(c + di) = (ac − bd) + (ad + bc)i; modulus |a + bi| = √(a² + b²).

Percent change = (new − old)/old × 100%; successive percent changes multiply, so +20% then −20% yields a net −4%.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying multiplies the coefficients and adds the exponents.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac gives two real roots (>0), one double root (=0), or complex conjugate roots (<0).

Vertex of y = ax² + bx + c occurs at x = −b/(2a), with vertex y-value found by substitution.

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a respectively.

Remainder Theorem: dividing polynomial p(x) by (x − r) leaves remainder p(r); p(r) = 0 iff (x − r) is a factor.

Rational Root Theorem: any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

Exponential growth/decay: A = A₀(1 + r)^t for growth rate r per period; continuous model A = A₀e^(kt).

Solving absolute-value and radical equations requires checking for extraneous solutions introduced by squaring or sign removal.

Logarithm laws: log(ab) = log a + log b, log(a/b) = log a − log b, log(a^k) = k log a; log_b x = ln x / ln b.

A relation is a function iff each input has exactly one output; it passes the vertical line test, and is one-to-one iff it also passes the horizontal line test.

Inverse functions: f⁻¹ exists iff f is one-to-one on its domain; the graph of f⁻¹ is the reflection of f across y = x, and (f⁻¹∘f)(x) = x.

Transformations of y = f(x): af(b(x − h)) + k gives vertical stretch |a|, horizontal compression 1/|b|, shift right h, shift up k.

Even functions satisfy f(−x) = f(x) (symmetric about the y-axis); odd functions satisfy f(−x) = −f(x) (symmetric about the origin).

Domain of f(x) = √g(x) requires g(x) ≥ 0; domain of 1/g(x) requires g(x) ≠ 0; domain of log g(x) requires g(x) > 0.

Composition (f∘g)(x) = f(g(x)) is generally not commutative: f∘g ≠ g∘f in most cases.

Exponential f(x) = b^x (b > 0, b ≠ 1) has domain all reals, range (0, ∞), and horizontal asymptote y = 0; its inverse is log_b x.

Rational function end behavior and asymptotes: vertical where denominator = 0 (after canceling); horizontal determined by comparing degrees of numerator and denominator.

Areas: triangle ½bh, trapezoid ½(b₁ + b₂)h, circle πr², sector ½r²θ with θ in radians.

Volume: prism/cylinder Bh, pyramid/cone (1/3)Bh, sphere (4/3)πr³; surface area of sphere 4πr².

Pythagorean theorem a² + b² = c²; common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Similar figures: corresponding sides scale by k, areas by k², volumes by k³.

Interior angle sum of an n-gon is (n − 2)·180°; each interior angle of a regular n-gon is (n − 2)·180°/n.

Arc length s = rθ and sector area A = ½r²θ when θ is measured in radians.

Distance and midpoint: d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Coordinate geometry: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ; also tan θ = sin θ/cos θ and cot θ = cos θ/sin θ.

Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.

Sum formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

Double-angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.

Law of Sines: a/sin A = b/sin B = c/sin C; Law of Cosines: c² = a² + b² − 2ab cos C.

Law of Cosines determines a triangle uniquely (SSS, SAS); Law of Sines has the ambiguous SSA case producing 0, 1, or 2 triangles.

Radian-degree conversion: 180° = π radians; arc length and angular speed use radians, with linear speed v = rω.

Amplitude and period of y = A sin(Bx + C) + D: amplitude |A|, period 2π/|B|, phase shift −C/B, vertical shift D.

Mean = sum/n; median is the middle value of ordered data; mode is the most frequent value; the mean is pulled toward skew and outliers.

Standard deviation measures spread about the mean; roughly 68% of normal data lie within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

z-score = (x − μ)/σ expresses how many standard deviations a value lies from the mean.

Permutations P(n, r) = n!/(n − r)! count ordered arrangements; combinations C(n, r) = n!/[r!(n − r)!] count unordered selections.

Probability of A or B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); independent events satisfy P(A ∩ B) = P(A)P(B).

Conditional probability P(A|B) = P(A ∩ B)/P(B); A and B are independent iff P(A|B) = P(A).

Binomial probability P(X = k) = C(n, k)p^k(1 − p)^(n − k) for n independent trials with success probability p.

Correlation coefficient r ranges from −1 to 1; r measures only the strength of linear association, not causation.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h.

Derivative rules: power d/dx xⁿ = nx^(n−1); product (fg)′ = f′g + fg′; quotient (f/g)′ = (f′g − fg′)/g²; chain d/dx f(g(x)) = f′(g(x))g′(x).

Derivatives: d/dx sin x = cos x, d/dx cos x = −sin x, d/dx e^x = e^x, d/dx ln x = 1/x.

A function has a local extremum where f′ = 0 or is undefined; the sign change of f′ (or sign of f″) classifies max versus min.

Definite integral as net signed area: ∫ₐᵇ f(x) dx = F(b) − F(a) by the Fundamental Theorem of Calculus.

Sequences and series: arithmetic aₙ = a₁ + (n − 1)d with sum n(a₁ + aₙ)/2; geometric aₙ = a₁r^(n−1) with sum a₁(1 − rⁿ)/(1 − r).

Infinite geometric series converges to a₁/(1 − r) iff |r| < 1.

Mathematical induction: prove the base case, assume the statement for n = k, then derive it for n = k + 1 to conclude for all positive integers n.