MB Pre-Calculus 40S cheat sheet

For y = a·f(b(x − h)) + k, the mapping from (x, y) on y = f(x) to the image point is (x/b + h, a·y + k).

Vertical stretch by |a|; reflection in the x-axis when a < 0. Horizontal stretch by 1/|b|; reflection in the y-axis when b < 0.

Translations are applied last: horizontal shift h units (right if h > 0), vertical shift k units (up if k > 0).

Even function satisfies f(−x) = f(x) and is symmetric about the y-axis; odd function satisfies f(−x) = −f(x) and is symmetric about the origin.

Inverse relation is found by interchanging x and y; the graph of f⁻¹ is the reflection of f in the line y = x.

A relation is a function only if it passes the vertical line test; its inverse is a function only if the original passes both vertical and horizontal line tests.

Reciprocal transformation y = 1/f(x): zeros of f become vertical asymptotes, and invariant points occur where f(x) = 1 or f(x) = −1.

Absolute value transformation y = |f(x)| reflects every part of the graph below the x-axis up above it, keeping parts above unchanged.

For y = a√(b(x − h)) + k, the domain requires b(x − h) ≥ 0; the endpoint of the graph is at (h, k).

To solve a radical equation, isolate the radical, square both sides, solve, then check every root in the original equation to reject extraneous solutions.

Rational function y = P(x)/Q(x): vertical asymptotes where Q(x) = 0 and P(x) ≠ 0; a hole where a common factor cancels.

Horizontal asymptote rules: degree P < degree Q gives y = 0; equal degrees give y = (leading coefficient ratio); degree P > degree Q gives none (unless a slant case).

If degree of numerator exceeds denominator by exactly one, the graph has an oblique (slant) asymptote found by division.

x-intercepts of a rational function occur at zeros of the numerator that are not also zeros of the denominator.

Solving a rational equation: multiply by the lowest common denominator, solve, then exclude any value that makes a denominator zero.

A radical function's range for even roots is y ≥ k when a > 0 and y ≤ k when a < 0.

Arc length s = rθ and sector area A = ½r²θ, with θ measured in radians; convert degrees to radians by multiplying by π/180.

Unit circle: (cos θ, sin θ) = (x, y); Pythagorean identity sin²θ + cos²θ = 1, with tan θ = sin θ / cos θ.

Exact values: sin 30° = ½, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2; use reference angles in each quadrant.

CAST rule: All ratios positive in quadrant I, Sine in II, Tangent in III, Cosine in IV.

For y = a sin b(x − c) + d: amplitude |a|, period 2π/|b|, phase shift c, vertical displacement d.

General solution pattern: if sin θ = k, then θ = θ_ref + 2πn or θ = π − θ_ref + 2πn for integer n.

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

The ambiguous case (given SSA with the sine law) can yield 0, 1, or 2 triangles; check whether the supplementary angle is valid.

Laws of logarithms: log(MN) = log M + log N, log(M/N) = log M − log N, log(M^p) = p·log M.

Change of base: log_b M = log M / log b (any common base), used to evaluate logs on a calculator.

Exponential equation solving: rewrite both sides with the same base, or take the logarithm of both sides.

Logarithmic equation solving: condense to a single log, convert to exponential form, then reject solutions that make any argument non-positive.

y = a·b^x has y-intercept a; growth when b > 1, decay when 0 < b < 1; horizontal asymptote y = 0.

y = log_b x has domain x > 0, range all reals, vertical asymptote x = 0, and x-intercept 1.

Half-life model A = A₀(½)^(t/h) and compound interest A = P(1 + r/n)^(nt); continuous growth A = Pe^(rt).

log_b b = 1, log_b 1 = 0, and b^(log_b x) = x for x > 0.

Fundamental counting principle: if tasks can be done in m and n ways, together they can be done in m × n ways.

Permutation nPr = n!/(n − r)! counts ordered arrangements; combination nCr = n!/(r!(n − r)!) counts unordered selections.

Arrangements of n objects with repeats: n!/(a!·b!·…), where a, b are the counts of identical items.

Circular permutations of n distinct objects: (n − 1)! arrangements around a circle.

Binomial theorem: (x + y)^n = Σ from k = 0 to n of nCk · x^(n−k) · y^k.

General term of (x + y)^n is t_(k+1) = nCk · x^(n−k) · y^k; the k index starts at 0.

Pascal's triangle: each entry is the sum of the two above it; row n gives the coefficients of (x + y)^n.

nC0 = nCn = 1 and nCk = nC(n−k), a symmetry used to simplify computations.

Circle: (x − h)² + (y − k)² = r², centre (h, k), radius r.

Parabola: (y − k)² = 4p(x − h) opens left/right; (x − h)² = 4p(y − k) opens up/down; vertex (h, k), focal length |p|.

Ellipse: (x − h)²/a² + (y − k)²/b² = 1; major axis length 2a, minor axis 2b, and c² = a² − b² for foci.

Hyperbola: (x − h)²/a² − (y − k)²/b² = 1 opens left/right; asymptotes y − k = ±(b/a)(x − h).

For a hyperbola, c² = a² + b²; the foci lie along the transverse axis.

General second-degree form Ax² + Cy² + Dx + Ey + F = 0 is a circle or ellipse when A and C have the same sign, a hyperbola when signs differ, a parabola when A or C is zero.

Completing the square converts general form to standard form, revealing centre, vertices, and axes.

Distance between foci of an ellipse is 2c; eccentricity e = c/a is less than 1 for ellipses and greater than 1 for hyperbolas.

Arithmetic sequence: t_n = t₁ + (n − 1)d; arithmetic series S_n = (n/2)(t₁ + t_n).

Geometric sequence: t_n = t₁·r^(n−1); geometric series S_n = t₁(r^n − 1)/(r − 1) for r ≠ 1.

Infinite geometric series converges to S = t₁/(1 − r) only when |r| < 1; otherwise it diverges.

Sigma notation: Σ from k = 1 to n of a_k means a₁ + a₂ + … + a_n; index shifts change the starting value but not the sum.

Sum of the first n natural numbers: Σ k = n(n + 1)/2; sum of squares Σ k² = n(n + 1)(2n + 1)/6.

Recursive definitions state t₁ and a rule such as t_n = t_(n−1) + d; explicit formulas give t_n directly.

To find n in a series problem, solve the term or sum formula for n and reject non-integer or non-positive results.

Arithmetic means between two numbers a and b: insert m means so the sequence has common difference d = (b − a)/(m + 1).