A formula booklet is supplied in every paper, so this sheet concentrates on the definitions, conditions and ideas you must reason out for yourself, restating a few results for quick recall. Learn what each symbol means and when you would choose the idea. HL extras are at the end.
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(a\times10^{k},\ 1\le a<10,\ k\in\mathbb Z\) | Standard form: a is the coefficient and k the power of ten | Writing very large or very small values |
| \(\text{percentage change}=\dfrac{\text{new}-\text{old}}{\text{old}}\times100\) | Positive for an increase, negative for a decrease | Comparing a value before and after |
| \(\text{growth multiplier}=1+\dfrac{r}{100}\) | r is the percentage increase per period | Setting up exponential growth |
| \(\text{decay multiplier}=1-\dfrac{r}{100}\) | r is the percentage decrease per period | Depreciation and decay |
| \(\text{real value}=\dfrac{\text{nominal value}}{(1+i)^{n}}\) | i is the yearly inflation rate as a decimal, n the number of years | Comparing money across years |
| \(k\text{ compounding periods: rate per period}=\dfrac{r}{k}\) | r is the annual nominal rate, k the number of periods per year | Entering finance values into the GDC |
| \(\text{3 s.f. unless told otherwise}\) | Keep full GDC values in working and round only the final answer | Every numerical answer |
| \(\text{error}=\text{approximate}-\text{exact}\) | Sign shows over or under-estimate | Judging accuracy of an estimate |
| \(\text{growth multiplier}=1\) | A percentage rate of zero, so the quantity does not change | Reading a growth model |
| \(\text{doubling: }(1+i)^n=2\) | Solve for n on the GDC | Time to double an investment |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{gradient}=\dfrac{\Delta y}{\Delta x}\) | Rate of change of y with respect to x, with units y per x | Interpreting a linear model |
| \(c\text{ is the value of }y\text{ when }x=0\) | y-intercept; give it a meaning in context | Reading starting values |
| \(\text{sinusoidal: amplitude}=\dfrac{\max-\min}{2}\) | Half the vertical distance between peak and trough | Tide and temperature models |
| \(\text{midline}=\dfrac{\max+\min}{2}\) | Value halfway between the extremes | Tide and temperature models |
| \(\text{period}=\dfrac{360^\circ}{b}\ \text{or}\ \dfrac{2\pi}{b}\) | b is the coefficient of x inside the sine or cosine | Matching a model to a cycle length |
| \(y=ab^{x}\) | Exponential model with initial value a and multiplier b | Growth and decay data |
| \(y=ax^{n}\) | Power model. At HL, plot ln y against ln x to see a straight line | Scaling relationships |
| \(y=\dfrac{a}{x}\) | Inverse variation, with axes as asymptotes | Quantities that fall as the other rises |
| (HL) \(\text{logistic: }f(x)=\dfrac{L}{1+Ce^{-kx}}\) | L is the limiting value; growth levels off | Populations with a carrying capacity |
| \(\text{domain}\ /\ \text{range}\) | The valid inputs and resulting outputs for the context | Stating limits of a model |
| (HL) \(R^{2}\text{ near }1\) | The fitted model explains most of the variation | Comparing regression models |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}\) | Right-angled triangle ratio | Finding sides and angles |
| \(\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}\) | Right-angled triangle ratio | Finding sides and angles |
| \(\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\) | Right-angled triangle ratio | Slopes and angles of elevation |
| \(a^{2}+b^{2}=c^{2}\) | c is the hypotenuse of a right-angled triangle | Distances in 2D and 3D |
| \(\text{area of rectangle}=lw\) | Length times width | Composite areas |
| \(\text{area of parallelogram}=bh\) | Base times perpendicular height | Composite areas |
| \(\text{area of trapezoid}=\dfrac{(a+b)}{2}h\) | a and b parallel sides, h the distance between them | Composite areas |
| \(\text{volume of prism}=\text{base area}\times\text{height}\) | Any solid with a constant cross-section | Containers and solids |
| \(\text{bearing: clockwise from north, 3 digits}\) | Measured from the north line at the starting point | Navigation problems |
| \(\text{angle of elevation = angle of depression (alternate angles)}\) | Equal angles between a horizontal line and a line of sight | Height and distance problems |
| \(\text{Voronoi edge: points equidistant from two sites}\) | Edges lie on perpendicular bisectors of site pairs | Nearest-site questions |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(P(A')=1-P(A)\) | Probability of the complement | Finding a probability by using its opposite |
| \(\text{independent}\iff P(A\mid B)=P(A)\) | Knowing B does not change the chance of A | Testing independence |
| \(\text{mutually exclusive}\iff P(A\cap B)=0\) | The events cannot both occur | Choosing between add and multiply rules |
| \(\text{expected count}=n\times p\) | n trials, probability p of the event | Predicting how many |
| \(\text{range}=\max-\min\) | Overall spread of the data | Comparing data sets |
| \(\text{IQR}=Q_3-Q_1\) | Spread of the middle half of the data | Comparing data sets |
| \(\text{class midpoint}=\dfrac{\text{lower}+\text{upper}}{2}\) | Used as the representative value of a group | Estimating a mean from grouped data |
| \(\text{reject }H_0\ \text{if}\ p<\alpha\) | p-value below the significance level | Any hypothesis test |
| \(H_0\text{ states no effect or no association}\) | The null hypothesis in a \(\chi^2\) test of independence says the variables are independent | Writing hypotheses |
| \(\text{correlation }r:\ -1\le r\le1\) | Sign gives direction; size gives strength | Interpreting a coefficient |
| \(\text{expected frequency}=\dfrac{\text{row total}\times\text{column total}}{\text{grand total}}\) | Cell value assuming independence | Setting up a \(\chi^2\) test |
| \(\text{degrees of freedom}=(r-1)(c-1)\) | r rows and c columns of the table | Choosing the \(\chi^2\) critical value |
| \(\text{normal: about 95\% within 2 standard deviations of the mean}\) | Rough rule for checking answers | Sanity checks |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(f'(x)\text{ is the gradient of the tangent}\) | Instantaneous rate of change | Interpreting a derivative |
| \(f'(x)=0\text{ at a stationary point}\) | Possible maximum, minimum or inflection | Optimisation |
| \(f'(x)>0\Rightarrow\text{increasing}\) | The function is rising where its derivative is positive | Sketching and interpretation |
| \(\text{normal gradient}=-\dfrac{1}{f'(a)}\) | Perpendicular to the tangent at x = a | Equation of a normal |
| \(\int_a^b f(x)\,dx\) | Area between the curve and the x-axis when f is positive | Accumulated change |
| \(\text{units of }\dfrac{dy}{dx}=\text{units of }y\text{ per unit of }x\) | Always state units when interpreting | Rates in context |
| (HL) \(\text{velocity}=\dfrac{ds}{dt}\) | Gradient of a displacement graph | Motion models |
| (HL) \(\text{total distance}=\int|v|\,dt\) | Use the absolute value of velocity | Total distance travelled |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{Euler's method: }y_{n+1}=y_n+h\,f(x_n,y_n)\) | h is the step size | Approximating differential equation solutions |
| \(\text{steady state: }\mathbf{M}\mathbf{s}=\mathbf{s}\) | The state vector does not change after one step | Markov chains |
| \(\text{Euler circuit exists}\iff\text{all vertex degrees even}\) | For a connected graph | Route planning |
| \(\sum\deg(v)=2\times\text{number of edges}\) | Handshaking relationship for graphs | Checking graph facts |
| \(\text{null hypothesis for }\mu\text{ test}\) | State H0 and H1 clearly in terms of the population value | Hypothesis tests |