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IB Math AA SL formulas not in the formula booklet

The exam gives you a formula booklet, but it leaves these 15 out. Learn them by heart. Each has a short worked example: tap “Worked example” to open it.

The IB gives this booklet to schools rather than posting it, so we checked the list against our own copy, not the IB's file. Your teacher has the official booklet. Official information: IB DP Mathematics page.

Test yourself with flashcards One-page formula sheet

Algebra and functions

Laws of indices

\(\displaystyle a^ma^n=a^{m+n},\) \(\displaystyle \frac{a^m}{a^n}=a^{m-n},\) \(\displaystyle (a^m)^n=a^{mn},\) \(\displaystyle a^{-n}=\frac1{a^n},\) \(\displaystyle a^{\frac mn}=\left(\sqrt[n]a\right)^m\)

Use it for: Simplifying powers and roots without a calculator, and as the first step of many log and calculus questions.

Worked example

Work out \(8^{\frac23}\times4^{-\frac12}\) without a calculator.

  1. \(8^{\frac23}=\left(\sqrt[3]8\right)^2=2^2=4\)
  2. \(4^{-\frac12}\)​\({}=\frac1{\sqrt4}\)​\({}=\frac12\)
  3. \(4\times\frac12=2\)

Answer: \(2\)

Perpendicular gradients

\(\displaystyle m_1m_2=-1\) \(\displaystyle \Big(m_2\)​\(\displaystyle {}=-\frac1{m_1}\Big)\)

Use it for: Finding a perpendicular line, a normal, or a perpendicular bisector.

Worked example

Line \(L\) passes through \((2,1)\) and meets \(y=3x-4\) at right angles. Give the equation of \(L\).

  1. The given gradient is \(3\), so the perpendicular gradient is \(-\frac13\).
  2. \(y-1=-\frac13(x-2)\)
  3. \(y=-\frac13x+\frac53\)

Answer: \(y=-\frac13x+\frac53\) (or \(x+3y-5=0\))

Vertex form of a quadratic

\(\displaystyle y=a(x-h)^2+k\ \text{ has vertex }(h,k)\)

Use it for: Finding a turning point, a range, or the transformations of \(y=x^2\).

Worked example

Write \(y=2x^2-12x+13\) in the form \(a(x-h)^2+k\) and state the vertex.

  1. \(2x^2-12x+13=2(x^2-6x)+13\)
  2. \(=2\big((x-3)^2-9\big)+13\)
  3. \(=2(x-3)^2-5\)

Answer: \(y=2(x-3)^2-5\), vertex \((3,-5)\)

Finding an inverse function

\(\displaystyle f\big(f^{-1}(x)\big)=x;\) \(\displaystyle \text{domain of }f^{-1}\)​\(\displaystyle {}=\text{range of }f\)

Use it for: Finding \(f^{-1}(x)\): write \(x=f(y)\) (swap \(x\) and \(y\)), then make \(y\) the subject.

Worked example

Find \(f^{-1}(x)\) for \(f(x)=e^{2x}+1\), \(x\in\mathbb R\), and state its domain.

  1. Write \(x=e^{2y}+1\).
  2. \(e^{2y}=x-1\)​\({}\Rightarrow 2y\)​\({}=\ln(x-1)\)
  3. The range of \(f\) is \(y>1\), so the domain of \(f^{-1}\) is \(x>1\).

Answer: \(f^{-1}(x)=\frac12\ln(x-1)\), \(x>1\)

Asymptotes of a rational function

\(\displaystyle y=\frac{ax+b}{cx+d}:\) \(\displaystyle x=-\frac dc,\) \(\displaystyle y=\frac ac\)

Use it for: Sketching reciprocal-type graphs and stating domains and ranges.

Worked example

State the asymptotes of \(y=\frac{2x+1}{x-3}\).

  1. Vertical: the denominator is zero when \(x=3\).
  2. Horizontal: as \(x\to\pm\infty\), \(y\to\frac21=2\).

Answer: \(x=3\) and \(y=2\)

Trigonometry

Exact trig values

\(\displaystyle \sin\tfrac\pi6=\tfrac12,\) \(\displaystyle \sin\tfrac\pi4\)​\(\displaystyle {}=\tfrac{\sqrt2}2,\) \(\displaystyle \sin\tfrac\pi3\)​\(\displaystyle {}=\tfrac{\sqrt3}2;\) \(\displaystyle \cos\text{ in reverse order};\) \(\displaystyle \tan\tfrac\pi6\)​\(\displaystyle {}=\tfrac1{\sqrt3},\ \tan\tfrac\pi4\)​\(\displaystyle {}=1,\ \tan\tfrac\pi3\)​\(\displaystyle {}=\sqrt3\)

Use it for: Every non-calculator trig question.

Worked example

Find the exact value of \(\sin\frac\pi3\cos\frac\pi6+\tan\frac\pi4\).

  1. \(\frac{\sqrt3}2\times\frac{\sqrt3}2\)​\({}=\frac34\)
  2. \(\frac34+1=\frac74\)

Answer: \(\frac74\)

The second solution of sin x = k

\(\displaystyle \sin x=k\ \)​\(\displaystyle {}\Rightarrow\ x\)​\(\displaystyle {}=\arcsin k\ \text{ or }\ \pi-\arcsin k\ \ (+2\pi n)\)

Use it for: Solving trig equations over an interval: the calculator gives only one answer.

Worked example

Find every \(x\) with \(0\le x\le2\pi\) and \(2\sin x=\sqrt3\).

  1. \(\sin x\)​\({}=\frac{\sqrt3}2\)​\({}\Rightarrow x\)​\({}=\arcsin\frac{\sqrt3}2\)​\({}=\frac\pi3\)
  2. Second solution: \(\pi-\frac\pi3\)​\({}=\frac{2\pi}3\)
  3. Adding \(2\pi\) takes both out of the interval.

Answer: \(x\)​\({}=\frac\pi3,\ \frac{2\pi}3\)

Amplitude and period

\(\displaystyle y=a\sin\big(b(x-c)\big)+d:\) \(\displaystyle \text{amplitude }|a|,\) \(\displaystyle \text{period }\frac{2\pi}b,\) \(\displaystyle \text{principal axis }y=d\)

Use it for: Matching a graph to its equation, and modelling questions.

Worked example

For \(y\)​\({}=3\sin\big(2(x-\frac\pi4)\big)+1\), find the amplitude, the period and the maximum value.

  1. Amplitude \(=|3|=3\)
  2. Period \(=\frac{2\pi}2=\pi\)
  3. Maximum \(=1+3=4\)

Answer: Amplitude 3, period \(\pi\), maximum 4

Statistics

Effect of scaling data

\(\displaystyle y=ax+b\ \)​\(\displaystyle {}\Rightarrow\ \bar y=a\bar x+b,\) \(\displaystyle \sigma_y=|a|\,\sigma_x\)

Use it for: Questions that change units or scale marks.

Worked example

Test marks have mean 52 and standard deviation 8. Each mark is scaled by \(y=1.5x+10\). Find the new mean and standard deviation.

  1. Mean: \(1.5\times52+10=88\)
  2. Standard deviation: \(1.5\times8=12\) (adding 10 does not change the spread)

Answer: Mean 88, standard deviation 12

Outliers

\(\displaystyle x \(\displaystyle \text{or}\) \(\displaystyle x>Q_3+1.5\times IQR\)

Use it for: Box plots and "is this value an outlier?" questions.

Worked example

A data set has \(Q_1=14\) and \(Q_3=22\). Is 36 an outlier?

  1. \(IQR=22-14=8\)
  2. Fences: \(14-12=2\) and \(22+12=34\)
  3. \(36>34\), so 36 is an outlier.

Answer: Yes: the fences are 2 and 34

Calculus

Tangent and normal at a point

\(\displaystyle \text{tangent: }y-f(a)=f'(a)(x-a);\) \(\displaystyle m_{\text{normal}}\)​\(\displaystyle {}=-\frac1{f'(a)}\)

Use it for: Any "find the equation of the tangent / normal" question.

Worked example

Find the tangent and the normal to \(y=x^3-2x\) at \(x=1\).

  1. \(y(1)=1-2=-1\)
  2. \(\frac{dy}{dx}=3x^2-2=1\) at \(x=1\)
  3. Tangent: \(y+1\)​\({}=1(x-1)\)​\({}\Rightarrow y=x-2\)
  4. Normal gradient \(-1\): \(y+1\)​\({}=-(x-1)\)​\({}\Rightarrow y\)​\({}=-x\)

Answer: Tangent \(y=x-2\), normal \(y=-x\)

Stationary points and their nature

\(\displaystyle f'(x)=0;\) \(\displaystyle f''(x)<0\)​\(\displaystyle {}\Rightarrow\text{max},\) \(\displaystyle f''(x)>0\)​\(\displaystyle {}\Rightarrow\text{min}\)

Use it for: Optimisation and curve sketching.

Worked example

Locate the stationary points of \(f(x)=x^3-9x^2+15x+4\) and say which is a maximum.

  1. \(f'(x)\)​\({}=3x^2-18x+15\)​\({}=3(x-1)(x-5)\)​\({}=0\)​\({}\Rightarrow x\)​\({}=1,\ 5\)
  2. \(f''(x)=6x-18\): \(f''(1)=-12<0\), \(f''(5)=12>0\)
  3. \(f(1)=11\), \(f(5)=-21\)

Answer: Maximum \((1,11)\), minimum \((5,-21)\)

Integrating a function of ax + b

\(\displaystyle \int f(ax+b)\,dx\)​\(\displaystyle {}=\frac1aF(ax+b)+C\)

Use it for: Integrals like \((3x-1)^4\), \(e^{2x+1}\) or \(\cos(5x)\).

Worked example

Find \(\int(3x-1)^4\,dx\).

  1. Integrate the outside: \(\frac{(3x-1)^5}5\)
  2. Divide by the inside's \(x\)-coefficient, 3.

Answer: \(\frac{(3x-1)^5}{15}+C\)

Area between two curves

\(\displaystyle A\)​\(\displaystyle {}=\int_a^b\big|f(x)-g(x)\big|\,dx\)

Use it for: Shaded-region questions.

Worked example

Find the area enclosed by \(y=x^2\) and \(y=2x\).

  1. Intersections: \(x^2=2x\Rightarrow x=0,\ 2\)
  2. On \(0
  3. \(=\big[x^2-\frac{x^3}3\big]_0^2\)​\({}=4-\frac83\)

Answer: \(\frac43\)

Displacement vs distance

\(\displaystyle \text{displacement}\)​\(\displaystyle {}=\int_{t_1}^{t_2}v\,dt,\) \(\displaystyle \text{distance}\)​\(\displaystyle {}=\int_{t_1}^{t_2}|v|\,dt\)

Use it for: Kinematics questions where the particle changes direction.

Worked example

\(v(t)=t^2-4\) for \(0\le t\le3\). Find the displacement and the distance travelled.

  1. Displacement: \(\int_0^3(t^2-4)\,dt\)​\({}=9-12\)​\({}=-3\)
  2. \(v<0\) on \(0
  3. Distance \(=\frac{16}3+\frac73\)​\({}=\frac{23}3\)

Answer: Displacement \(-3\), distance \(\frac{23}3\)

Keep going

Questions

Which IB Math AA SL formulas are not in the formula booklet?

The ones to learn by heart are: Laws of indices; Perpendicular gradients; Vertex form of a quadratic; Finding an inverse function; Asymptotes of a rational function; Exact trig values; The second solution of sin x = k; Amplitude and period; Effect of scaling data; Outliers; Tangent and normal at a point; Stationary points and their nature; Integrating a function of ax + b; Area between two curves; Displacement vs distance. Each one is on this page with a worked example.

Do I get a formula booklet in the exam?

Yes. You get a clean copy of the IB formula booklet in every IB Mathematics exam, so you only need to memorise what it leaves out.

What is the best way to memorise them?

Cover the formula, write it from memory, then try the worked example without looking. Come back to the ones you missed the next day. Our flashcards do the spacing for you.

Is this page a copy of the official booklet?

No. We link to the official IB Mathematics: analysis and approaches formula booklet; we do not reproduce it. This page lists only what the booklet leaves out, in our own words, with our own examples.

Our own list, wording and examples, written and checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.