MHF4U Grade 12 Advanced Functions — Exam Cheat Sheet

Division Algorithm: P(x) = d(x) · Q(x) + R(x) where degree(R) < degree(d).

Remainder Theorem: When a polynomial P(x) is divided by (ax - b), the remainder is R = P(b/a).

Factor Theorem: (ax - b) is a factor of P(x) if and only if P(b/a) = 0.

Rational Root Theorem: Possible rational roots of P(x) are of the form ±(factors of constant term) / (factors of leading coefficient).

Vertical Asymptote (VA): Occurs where the denominator equals zero and factors do NOT cancel with the numerator: x = c.

Hole (Removable Discontinuity): Occurs where identical factors cancel in both numerator and denominator.

Horizontal Asymptote (HA) for f(x) = P(x) / Q(x):

• degree(P) < degree(Q): HA is always y = 0.
• degree(P) = degree(Q): HA is ratio of leading coefficients: y = a_n / b_n.
• degree(P) = degree(Q) + 1: No horizontal asymptote; an oblique/slant asymptote exists (found via polynomial long division quotient).

Radian Conversion: π radians = 180°; Radians = Degrees · (π/180); Arc length: a = r · θ

Addition / Subtraction Formulas:

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Double Angle Formulas:

• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
• tan(2θ) = (2 tan θ) / (1 - tan² θ)

Equivalence: y = log_b(x) ⟺ b^y = x (where b > 0, b ≠ 1, x > 0)

Product Law: log_b(xy) = log_b(x) + log_b(y)

Quotient Law: log_b(x/y) = log_b(x) - log_b(y)

Power Law: log_b(x^k) = k · log_b(x)

Change of Base: log_b(x) = log(x) / log(b) = ln(x) / ln(b)