HSC Maths Advanced cheat sheet

A relation is a function if every x-value maps to exactly one y-value; the vertical line test detects failures.

Domain restrictions: exclude values making a denominator zero, and require the radicand of an even root to be non-negative.

Composite function (f∘g)(x)=f(g(x)) exists only where x is in the domain of g and g(x) is in the domain of f.

Inverse function f⁻¹ exists only if f is one-to-one; its graph is the reflection of y=f(x) in the line y=x, and domain and range swap.

Transformations of y=f(x): y=f(x−h)+k shifts right h and up k; y=af(x) dilates vertically by |a|; y=f(bx) dilates horizontally by 1/|b|.

Odd functions satisfy f(−x)=−f(x) (rotational symmetry about the origin); even functions satisfy f(−x)=f(x) (symmetry about the y-axis).

Absolute value: |x| equals x when x≥0 and −x when x<0; solve |expression|=k by splitting into expression=k and expression=−k, then check solutions.

For the quadratic y=ax²+bx+c, the axis of symmetry is x=−b/(2a) and the discriminant Δ=b²−4ac gives two, one or no real roots as Δ>0, =0, <0.

For a right triangle: sin θ=opposite/hypotenuse, cos θ=adjacent/hypotenuse, tan θ=opposite/adjacent; also tan θ=sin θ/cos θ.

Exact values: sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3; sin 45°=cos 45°=1/√2; sin 60°=√3/2, cos 60°=1/2, tan 60°=√3.

ASTC rule: in quadrants I, II, III, IV respectively, All, Sine, Tangent, Cosine are positive; the related acute angle gives the second solution.

Identity sin²θ+cos²θ=1 converts between sine and cosine; dividing it by cos²θ gives 1+tan²θ=sec²θ.

Sine and cosine functions have period 360° (2π) and amplitude 1; tangent has period 180° (π) and no amplitude.

For y=a sin(bx+c)+d: amplitude=|a|, period=360°/|b|, phase shift=−c/b, vertical shift d.

Sine rule: a/sin A=b/sin B=c/sin C; use it when given a matching side-angle pair.

Cosine rule: c²=a²+b²−2ab cos C; area of a triangle=½ab sin C.

Radians: π radians=180°; arc length l=rθ and sector area A=½r²θ when θ is in radians.

Derivative from first principles: f′(x)=lim(h→0)[f(x+h)−f(x)]/h.

Power rule: d/dx(xⁿ)=nxⁿ⁻¹ for all real n; derivative of a constant is 0.

d/dx(eˣ)=eˣ; d/dx(ln x)=1/x for x>0; d/dx(sin x)=cos x and d/dx(cos x)=−sin x when x is in radians.

Product rule: (uv)′=u′v+uv′; quotient rule: (u/v)′=(u′v−uv′)/v²; chain rule: d/dx f(g(x))=f′(g(x))·g′(x).

The derivative is the gradient of the tangent; the normal at that point has gradient −1/m where m is the tangent gradient.

Stationary points occur where f′(x)=0; use the sign of f″ (or a sign table) to classify as maximum, minimum or horizontal inflection.

Increasing where f′(x)>0, decreasing where f′(x)<0; f″>0 indicates concavity upwards, f″<0 concavity downwards.

Integration reverses differentiation: ∫xⁿ dx=xⁿ⁺¹/(n+1)+C for n≠−1, and ∫(1/x) dx=ln|x|+C.

Definite integral: ∫ₐᵇ f(x) dx=F(b)−F(a); it gives the signed area between the curve and the x-axis, so areas below the axis count as negative.

Trapezoidal rule: ∫ₐᵇ f(x) dx≈(h/2)[f(x₀)+2f(x₁)+…+2f(xₙ₋₁)+f(xₙ)] with h=(b−a)/n.

Compound interest: A=P(1+r)ⁿ, where r is the decimal rate per period and n the number of periods; solve for any unknown by rearranging or using logarithms.

Depreciation: reducing-balance value S=V(1−r)ⁿ; straight-line depreciation subtracts a fixed amount D each period.

Future value of an annuity: FV=M[(1+r)ⁿ−1]/r, where M is the regular payment per period and r the rate per period.

Present value of an annuity: PV=M[1−(1+r)⁻ⁿ]/r; this is the lump sum equivalent today of n future payments.

For a reducing-balance loan, interest is charged on the outstanding balance each period before the repayment is subtracted.

Nominal annual rate i compounded m times per year gives periodic rate r=i/m and effective annual rate (1+i/m)^m−1.

Total repayment on a loan equals the number of payments times the payment amount; total interest is total repayment minus principal borrowed.

A recurrence relation of the form Aₙ₊₁=(1+r)Aₙ−M models a loan or annuity, with Aₙ the balance after n periods.

Mean x̄=Σx/n; median is the middle value of ordered data; mode is the most frequent value.

For grouped data, class centre is used as the representative value, and the mean is Σ(f·x)/Σf.

Range=max−min; interquartile range IQR=Q₃−Q₁; an outlier lies below Q₁−1.5·IQR or above Q₃+1.5·IQR.

Standard deviation measures spread about the mean; about 68% of data lies within one standard deviation and about 95% within two for approximately normal data.

z-score z=(x−x̄)/s states how many standard deviations a value lies from the mean; |z|>2 is often treated as unusual.

Correlation coefficient r lies between −1 and 1; r near ±1 indicates a strong linear association, r near 0 a weak one.

A least-squares regression line has gradient b=r·(s_y/s_x) and passes through the point (x̄, ȳ).

Correlation does not imply causation; extrapolating a regression line well beyond the data range is unreliable.

Relative frequency of an event=number of favourable outcomes/total trials, and probabilities of complementary events sum to 1.