Math 30-1 cheat sheet

Function: every x in the domain is paired with exactly one y; the vertical line test confirms this on a graph, since a line crossing twice would give one x two outputs.

Transformations of y = f(x): y = a·f(b(x − h)) + k maps each point (x, y) to (x/b + h, ay + k); |a| > 1 stretches vertically, 0 < |a| < 1 compresses, and a < 0 reflects in the x-axis.

Domain restrictions: exclude x-values making a denominator zero; require an even-root radicand to be ≥ 0; require a logarithm argument to be > 0.

Composition: (f ∘ g)(x) = f(g(x)); its domain excludes x not in the domain of g and x for which g(x) is not in the domain of f.

Inverse: interchange x and y and solve; an inverse function exists only when the original is one-to-one (horizontal line test), and the two graphs reflect in y = x.

Absolute value: |x| = x for x ≥ 0 and −x for x < 0; the graph of y = |f(x)| reflects every portion of y = f(x) lying below the x-axis above it.

Square root relation: y = √(f(x)) requires f(x) ≥ 0 and has range y ≥ 0; it is the inverse relation of y = x² restricted to x ≥ 0.

Rational functions: a non-cancelling factor (x − c) in the denominator gives vertical asymptote x = c; if numerator degree < denominator degree the horizontal asymptote is y = 0, and if degrees are equal it is y = (ratio of leading coefficients).

Polynomials: degree n has at most n x-intercepts and n − 1 turning points; even degree gives the same end behaviour on both sides, odd degree gives opposite end behaviour.

Radian measure: θ = s/r, where s is arc length and r is radius; π radians = 180°, so 1 radian ≈ 57.3°.

Unit circle: the terminal point for angle θ is (cos θ, sin θ); tan θ = sin θ/cos θ and is undefined where cos θ = 0.

Pythagorean identities: sin²θ + cos²θ = 1; dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.

Exact values: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2; cosine values reverse this order, and each tangent is sine divided by cosine.

Signs by quadrant (ASTC): all ratios positive in quadrant I, sine in II, tangent in III, cosine in IV; the reference angle supplies the magnitude.

Sinusoidal form: y = a·sin b(x − c) + d has amplitude |a|, period 360°/|b| (or 2π/|b| in radians), phase shift c, and vertical displacement d.

General solutions: sin θ = k gives θ = arcsin k + 360°n or θ = 180° − arcsin k + 360°n; cos θ = k gives θ = ±arccos k + 360°n, for integer n.

Law of sines: a/sin A = b/sin B = c/sin C; the SSA (ambiguous) case can produce zero, one, or two valid triangles.

Law of cosines: c² = a² + b² − 2ab·cos C; use it for SAS and SSS triangles, and it reduces to the Pythagorean theorem when C = 90°.

Exponent laws: aᵐ·aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ; a⁻ⁿ = 1/aⁿ; a^(m/n) = ⁿ√(aᵐ).

Exponential function y = a·bˣ has y-intercept a, represents growth when b > 1 and decay when 0 < b < 1, and has horizontal asymptote y = 0.

Logarithm definition: log_b x = y exactly when bʸ = x, with b > 0, b ≠ 1, and x > 0; consequently log_b 1 = 0 and log_b b = 1.

Log laws: log_b(MN) = log_b M + log_b N; log_b(M/N) = log_b M − log_b N; log_b(Mᵖ) = p·log_b M, for positive M and N.

Change of base: log_b x = log_c x / log_c b; also b^(log_b x) = x and log_b(bˣ) = x, so exponential and logarithmic functions undo each other.

Solving aˣ = b: rewrite both sides with a common base and equate exponents, or take logarithms to obtain x = log b / log a.

Extraneous roots: after solving a logarithmic equation, reject any candidate that makes an original argument non-positive or produces an invalid base.

Inverse graphs: y = log_b x and y = bˣ reflect in y = x; the logarithmic graph has vertical asymptote x = 0 and passes through (1, 0) and (b, 1).

Growth and decay models: A = A₀·b^(t/p) for a quantity growing or decaying by factor b each period p; half-life uses A = A₀(1/2)^(t/h); compound interest uses A = P(1 + i)ⁿ.

Fundamental counting principle: if one stage can occur in m ways and a second in n ways, the two stages together occur in m·n ways; multiply across all stages.

Factorial: n! = n(n − 1)(n − 2)···(2)(1) for n ≥ 1, and 0! = 1 by definition; n! counts the orderings of n distinct objects.

Permutations (order matters): nPr = n!/(n − r)!, the number of ordered arrangements of r objects selected from n distinct objects without replacement.

Combinations (order does not matter): nCr = n!/(r!(n − r)!); also nCr = nC(n − r) and nC0 = nCn = 1.

Repetition and circles: arrangements of n objects with identical groups of sizes a, b, c, … number n!/(a!b!c!…); distinct objects arranged in a circle number (n − 1)!.

Case splitting: for 'at least' or 'at most' conditions, add the counts of all acceptable cases, or subtract the complement's count from the total.

Binomial theorem: (a + b)ⁿ = Σ (from r = 0 to n) nCr·a^(n−r)·b^r; the general term is t_(r+1) = nCr·a^(n−r)·b^r.

Binomial facts: the expansion has n + 1 terms, coefficients are symmetric, the sum of all coefficients is 2ⁿ, and row n of Pascal's triangle lists the coefficients nC0 through nCn.