Formula Summary

Key formulas for Cambridge IGCSE / O Level Additional Mathematics 0606, organised in coursebook chapter order.

1. Functions

Composite function: $fg(x) = f\bigl(g(x)\bigr)$
Inverse function: $f\bigl(f^{-1}(x)\bigr) = x = f^{-1}\bigl(f(x)\bigr)$
An inverse exists only when $f$ is one–one on its domain.
Domain of a composite: $x \in \mathrm{domain}(g)\quad\text{and}\quad g(x) \in \mathrm{domain}(f)$

Quadratic formula: $ax^{2}+bx+c=0 \ \Rightarrow\ x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
Discriminant: $\Delta = b^{2}-4ac$
$\Delta>0$ two real roots; $\Delta=0$ equal roots; $\Delta<0$ no real roots.
Completing the square: $x^{2}+bx = \left(x+\dfrac{b}{2}\right)^{2}-\left(\dfrac{b}{2}\right)^{2}$
Sum and product of roots: $\alpha+\beta=-\dfrac{b}{a},\qquad \alpha\beta=\dfrac{c}{a}$

Factor theorem: $f(a)=0 \ \Leftrightarrow\ (x-a)\text{ is a factor of }f(x)$
Remainder theorem: $f(x)=(x-a)q(x)+r \ \Rightarrow\ f(a)=r$

Modulus definition: $|x|=\begin{cases}x & x\ge 0\\ -x & x<0\end{cases}$
Squaring (non-negative sides): $|A|=|B| \ \Leftrightarrow\ A^{2}=B^{2}$
Linear modulus inequality: $|x-a|<k \ \Leftrightarrow\ a-k<x<a+k \quad (k>0)$

Product law: $\mathrm{log}_{a}(xy)=\mathrm{log}_{a}x+\mathrm{log}_{a}y$
Quotient law: $\mathrm{log}_{a}\!\left(\dfrac{x}{y}\right)=\mathrm{log}_{a}x-\mathrm{log}_{a}y$
Power law: $\mathrm{log}_{a}(x^{k})=k\,\mathrm{log}_{a}x$
Change of base: $\mathrm{log}_{a}x=\dfrac{\mathrm{log}_{b}x}{\mathrm{log}_{b}a},\qquad \mathrm{log}_{a}b=\dfrac{1}{\mathrm{log}_{b}a}$
Inverse relations: $\mathrm{log}_{a}(a^{x})=x,\qquad a^{\mathrm{log}_{a}x}=x$
Common / natural log: $\mathrm{lg}\,x=\mathrm{log}_{10}x,\qquad \mathrm{ln}\,x=\mathrm{log}_{e}x$
Solving $a^{x}=b$: $a^{x}=b \ \Rightarrow\ x=\dfrac{\mathrm{ln}\,b}{\mathrm{ln}\,a}=\mathrm{log}_{a}b$
Exponential growth / decay: $P=P_{0}\mathrm{e}^{kt}$
$k>0$ growth; $k<0$ decay. Doubling time $T=(\mathrm{ln}\,2)/k$ .
Domain: $\mathrm{log}_{a}x\text{ requires }x>0,\ a>0,\ a\neq 1$

Gradient: $m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$
Point–gradient form: $y-y_{1}=m(x-x_{1})$
Parallel / perpendicular: $\text{parallel: }m_{1}=m_{2},\qquad \text{perpendicular: }m_{1}m_{2}=-1$
Midpoint: $\left(\dfrac{x_{1}+x_{2}}{2},\,\dfrac{y_{1}+y_{2}}{2}\right)$
Distance: $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$
Linear law (example): $y=Ax^{b}\ \Rightarrow\ \mathrm{lg}\,y=\mathrm{lg}\,A+b\,\mathrm{lg}\,x$
Plot $\mathrm{lg}\,y$ against $\mathrm{lg}\,x$ : gradient $b$ , intercept $\mathrm{lg}\,A$ .

Circle equation: $(x-a)^{2}+(y-b)^{2}=r^{2}$
Centre $(a,b)$ , radius $r$ .
Expanded form: $x^{2}+y^{2}+2gx+2fy+c=0$
Centre $(-g,-f)$ , radius $\sqrt{g^{2}+f^{2}-c}$ .
Angle in a semicircle: $\text{angle in a semicircle is a right angle}$
Radius $\perp$ tangent: $\text{radius to the point of contact is perpendicular to the tangent}$

Pythagorean identities: $\mathrm{sin}^{2}\theta+\mathrm{cos}^{2}\theta=1,\quad 1+\mathrm{tan}^{2}\theta=\mathrm{sec}^{2}\theta,\quad 1+\mathrm{cot}^{2}\theta=\mathrm{cosec}^{2}\theta$
Double angle: $\mathrm{sin}\,2A=2\mathrm{sin}\,A\,\mathrm{cos}\,A$
Cosine double angle: $\mathrm{cos}\,2A=\mathrm{cos}^{2}A-\mathrm{sin}^{2}A=2\mathrm{cos}^{2}A-1=1-2\mathrm{sin}^{2}A$
Tangent double angle: $\mathrm{tan}\,2A=\dfrac{2\mathrm{tan}\,A}{1-\mathrm{tan}^{2}A}$
$a\mathrm{sin}\theta\pm b\mathrm{cos}\theta$: $R\,\mathrm{sin}(\theta\pm\alpha)\quad\text{or}\quad R\,\mathrm{cos}(\theta\mp\alpha)$
$R=\sqrt{a^{2}+b^{2}}$ ; choose $\alpha$ from $\mathrm{tan}\,\alpha=b/a$ (signs as appropriate).

AP $n$ th term: $u_{n}=a+(n-1)d$
AP sum: $S_{n}=\dfrac{n}{2}\bigl(2a+(n-1)d\bigr)=\dfrac{n}{2}(a+l)$
GP $n$ th term: $u_{n}=ar^{n-1}$
GP sum: $S_{n}=\dfrac{a(1-r^{n})}{1-r}\quad (r\neq 1)$
GP sum to infinity: $S_{\infty}=\dfrac{a}{1-r}\quad (|r|<1)$
Binomial expansion: $(a+b)^{n}=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k}$
For $|x|<1$ : $(1+x)^{n}=1+nx+\dfrac{n(n-1)}{2!}x^{2}+\cdots$

Power rule: $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(x^{n}\bigr)=nx^{n-1}$
Chain rule: $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\mathrm{d}y}{\mathrm{d}u}\,\dfrac{\mathrm{d}u}{\mathrm{d}x}$
Product rule: $\dfrac{\mathrm{d}}{\mathrm{d}x}(uv)=u\dfrac{\mathrm{d}v}{\mathrm{d}x}+v\dfrac{\mathrm{d}u}{\mathrm{d}x}$
Quotient rule: $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{u}{v}\right)=\dfrac{v\dfrac{\mathrm{d}u}{\mathrm{d}x}-u\dfrac{\mathrm{d}v}{\mathrm{d}x}}{v^{2}}$
Stationary points: $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$
Use the second-derivative test or a first-derivative sign check for nature.
Small increments: $\delta y \approx \dfrac{\mathrm{d}y}{\mathrm{d}x}\,\delta x$

Magnitude: $|\mathbf{a}|=\sqrt{a_{1}^{2}+a_{2}^{2}}$
Unit vector: $\hat{\mathbf{a}}=\dfrac{\mathbf{a}}{|\mathbf{a}|}$
Section formula: $\overrightarrow{OP}=\dfrac{n\,\mathbf{a}+m\,\mathbf{b}}{m+n}$
$P$ divides $AB$ in the ratio $m:n$ .
Parallel vectors: $\mathbf{a}=k\mathbf{b}\text{ for some scalar }k$

Exponential: $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(\mathrm{e}^{kx}\bigr)=k\mathrm{e}^{kx}$
Natural log: $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(\mathrm{ln}\,x\bigr)=\dfrac{1}{x}$
Sine / cosine: $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(\mathrm{sin}\,kx\bigr)=k\mathrm{cos}\,kx,\quad \dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(\mathrm{cos}\,kx\bigr)=-k\mathrm{sin}\,kx$
Tangent: $\dfrac{\mathrm{d}}{\mathrm{d}x}\bigl(\mathrm{tan}\,kx\bigr)=k\mathrm{sec}^{2}kx$

Power rule: $\int x^{n}\,\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+C\quad (n\neq -1)$
Reciprocal: $\int \dfrac{1}{x}\,\mathrm{d}x=\mathrm{ln}|x|+C$
Exponential: $\int \mathrm{e}^{kx}\,\mathrm{d}x=\dfrac{1}{k}\mathrm{e}^{kx}+C$
Sine / cosine: $\int \mathrm{sin}\,kx\,\mathrm{d}x=-\dfrac{1}{k}\mathrm{cos}\,kx+C,\quad \int \mathrm{cos}\,kx\,\mathrm{d}x=\dfrac{1}{k}\mathrm{sin}\,kx+C$
Area under a curve: $A=\int_{a}^{b}y\,\mathrm{d}x$
Take absolute values (or split) where the curve is below the axis.

Velocity / acceleration: $v=\dfrac{\mathrm{d}s}{\mathrm{d}t},\qquad a=\dfrac{\mathrm{d}v}{\mathrm{d}t}=\dfrac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}}$
Integrating motion: $s=\int v\,\mathrm{d}t,\qquad v=\int a\,\mathrm{d}t$
Useful form: $a=v\dfrac{\mathrm{d}v}{\mathrm{d}s}$

Multiplication / division: $a^{m}a^{n}=a^{m+n},\qquad \dfrac{a^{m}}{a^{n}}=a^{m-n}$
Power of a power: $(a^{m})^{n}=a^{mn}$
Negative / fractional: $a^{-n}=\dfrac{1}{a^{n}},\qquad a^{m/n}=\sqrt[n]{a^{m}}$
Rationalising (simple): $\dfrac{1}{\sqrt{a}}=\dfrac{\sqrt{a}}{a},\qquad \dfrac{1}{a+\sqrt{b}}=\dfrac{a-\sqrt{b}}{a^{2}-b}$