QCE Maths Methods cheat sheet

Quadratic formula: for ax²+bx+c=0 with a≠0, x = (−b ± √(b²−4ac))/(2a); the discriminant Δ=b²−4ac gives two real roots if Δ>0, one if Δ=0, none real if Δ<0.

Completing the square: ax²+bx+c = a(x+b/(2a))² + c − b²/(4a); the turning point of y=ax²+bx+c is at x=−b/(2a).

Factor theorem: (x−a) is a factor of P(x) exactly when P(a)=0; remainder theorem: dividing P(x) by (x−a) leaves remainder P(a).

Index laws: aᵐaⁿ=aᵐ⁺ⁿ, aᵐ/aⁿ=aᵐ⁻ⁿ, (aᵐ)ⁿ=aᵐⁿ, a⁰=1, a⁻ⁿ=1/aⁿ, and a^(m/n)=ⁿ√(aᵐ) for a>0.

Log laws: logₐ(xy)=logₐx+logₐy, logₐ(x/y)=logₐx−logₐy, logₐ(xⁿ)=n logₐx, logₐ1=0, logₐa=1; change of base logₐx = log_b x / log_b a.

Solving exponential equations: rewrite both sides with the same base, or take logs of both sides; e.g. aˣ=k gives x = logₐk = ln k / ln a.

Simultaneous equations: for a linear and a quadratic, substitute the linear relation into the quadratic to obtain a quadratic in one variable; the number of intersection points equals the number of real solutions.

Absolute value: |x| is the distance from x to 0; |x−a|=b (b≥0) means x=a+b or x=a−b, and |x−a|<b means a−b<x<a+b.

Function notation: f: D→R maps each x in the domain D to exactly one value f(x); a relation is a function only if it passes the vertical line test.

Transformations of y=f(x): y=af(b(x−h))+k gives vertical dilation by a, horizontal dilation by 1/b, translation h right and k up; dilations apply before translations.

Inverse functions exist only for one-to-one functions; the graph of f⁻¹ is the reflection of y=f(x) in the line y=x, and domain and range swap.

Composite functions: f(g(x)) is defined only for x in the domain of g with g(x) in the domain of f; in general f(g(x)) ≠ g(f(x)).

Odd and even symmetry: f is even if f(−x)=f(x) (symmetric about the y-axis), odd if f(−x)=−f(x) (rotational symmetry about the origin).

Asymptotes: a rational function has a vertical asymptote where the denominator is zero (unless the factor cancels), and a horizontal asymptote found by comparing degrees of numerator and denominator.

Quadratic graph features: y=a(x−h)²+k has turning point (h,k), axis of symmetry x=h; if a>0 it opens upward with a minimum, if a<0 downward with a maximum.

Sign diagrams: mark zeros and undefined points on a number line, test one value per interval, and record the sign of the expression to solve inequalities such as f(x)>0.

Derivative from first principles: f′(x) = lim(h→0) [f(x+h)−f(x)]/h; the derivative gives the gradient of the tangent at x.

Power rule: d/dx(xⁿ)=nxⁿ⁻¹ for all real n; also d/dx(constant)=0 and d/dx(cf(x))=c f′(x).

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

Exponential and log derivatives: d/dx(eˣ)=eˣ, d/dx(e^{kx})=k e^{kx}, d/dx(ln x)=1/x, d/dx(ln(kx))=1/x for x>0.

Trigonometric derivatives: d/dx(sin x)=cos x, d/dx(cos x)=−sin x, d/dx(tan x)=sec²x, with arguments in radians; chain rule gives d/dx sin(kx)=k cos(kx).

Stationary points occur where f′(x)=0; use the sign of f′(x) either side, or f″(x) (positive gives a local minimum, negative a local maximum) to classify.

Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1)+C for n≠−1, ∫1/x dx = ln|x|+C, ∫e^{kx} dx = e^{kx}/k + C, ∫cos x dx = sin x + C.

Definite integrals: ∫ₐᵇ f(x)dx = F(b)−F(a) gives signed area; the area between curves is ∫ₐᵇ |f(x)−g(x)| dx, and the area between a curve and the x-axis uses |f(x)|.

Discrete random variable: probabilities satisfy 0≤P(X=x)≤1 and ΣP(X=x)=1; the mean is μ=E(X)=Σx·P(X=x).

Variance: Var(X)=E(X²)−[E(X)]² = Σx²P(X=x) − μ²; the standard deviation is σ=√Var(X).

Binomial distribution: X~Bin(n,p) has P(X=x)=C(n,x)pˣ(1−p)ⁿ⁻ˣ, mean np, variance np(1−p); conditions are a fixed number n of independent trials each with the same success probability p.

Normal distribution: X~N(μ,σ²); probabilities are found from standardised values z=(x−μ)/σ using the standard normal table or technology, and the total area under the curve is 1.

Empirical rule for normal data: about 68% of values lie within μ±σ, about 95% within μ±2σ, and about 99.7% within μ±3σ.

Sample statistics: mean x̄=Σx/n; sample standard deviation s uses divisor (n−1); the median is the middle value of ordered data and is resistant to outliers.

Association between two variables is described by direction, form and strength; the correlation coefficient r satisfies −1≤r≤1, with r near ±1 indicating a strong linear association.

Least-squares regression line: ŷ=a+bx with b = r·(s_y/s_x), passing through the point (x̄, ȳ); r² is the proportion of variation in y explained by the linear model.

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

Unit circle definitions: cos θ and sin θ are the x- and y-coordinates of the point at angle θ; tan θ = sin θ/cos θ, undefined where cos θ=0.

Exact values: sin30°=cos60°=1/2, sin45°=cos45°=√2/2, sin60°=cos30°=√3/2, tan45°=1, and sin0=0, cos0=1.

Symmetry identities: sin(π−θ)=sin θ, cos(π−θ)=−cos θ, sin(θ+π)=−sin θ, cos(θ+π)=−cos θ, sin(−θ)=−sin θ, cos(−θ)=cos θ.

Pythagorean identity: sin²θ+cos²θ=1, giving tan²θ+1=sec²θ and 1+cot²θ=cosec²θ.

General solutions: sin θ=k gives θ=α+2nπ or θ=π−α+2nπ; cos θ=k gives θ=±α+2nπ, where α is the related base angle and n is an integer.

Graphs of y=a sin(b(x−c))+d: amplitude |a|, period 2π/b, phase shift c, vertical shift d; for y=a tan(bx) the period is π/b and there is no amplitude.

Compound and double angle formulas: sin(A±B)=sinA cosB ± cosA sinB; cos(A±B)=cosA cosB ∓ sinA sinB; sin2A=2 sinA cosA; cos2A=cos²A−sin²A.

Exponential function y=aˣ (a>0, a≠1) has domain R, range (0,∞), y-intercept 1, horizontal asymptote y=0; it is increasing if a>1 and decreasing if 0<a<1.

The number e≈2.71828 is the base for which d/dx(eˣ)=eˣ; the natural logarithm ln x is log_e x, so ln(eˣ)=x and e^{ln x}=x for x>0.

Logarithmic function y=logₐx is the inverse of y=aˣ: domain (0,∞), range R, vertical asymptote x=0, x-intercept 1.

Exponential growth and decay: y=Ae^{kt} grows if k>0 and decays if k<0, with A the initial value; the doubling time is ln2/k for growth and the half-life is ln2/|k| for decay.

Solving equations: aˣ=b gives x=logₐb; e^{kx}=c gives x=(ln c)/k; logₐ(x)=logₐ(y) implies x=y provided both are positive.

Logarithmic scales: a one-unit increase in log₁₀y corresponds to multiplication of y by 10, which underlies decibel and Richter-scale comparisons.

Graph transformations apply to exponentials and logs as for other functions: y=e^{x−h}+k shifts the asymptote to y=k, and y=ln(x−h) shifts the vertical asymptote to x=h.

Domain restrictions: ln(ax+b) requires ax+b>0, and log expressions in equations must be checked against the original domain to reject invalid solutions.