formula.expertYour record

Grade 12 Math: formula sheet

Rates of change, vectors, data and money · 70 formulas · metric edition 1

The full study guide: modules, problems, answer key
Module 1 · Rates of Change
Math 12-1 Slope Between Two Points
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
Steepness of a line as rise over run between two points.
Math 12-2 Displacement from Average Velocity
d=v0+v2 td = \frac{v_0 + v}{2} \, t
Displacement as the average of initial and final velocities multiplied by the elapsed time, valid…
Math 12-3 Final Velocity (Uniform Acceleration)
v=v0+atv = v_0 + a t
Final velocity after accelerating uniformly from an initial velocity for a given time.
Math 12-4 Displacement (Uniform Acceleration)
d=v0t+12at2d = v_0 t + \tfrac{1}{2} a t^2
Distance travelled under constant acceleration, starting from an initial velocity, over a time t.
Math 12-5 Velocity-Displacement Relation (v² = v₀² + 2ad)
v2=v02+2adv^2 = v_0^2 + 2 a d
Links initial and final speeds to acceleration and displacement without involving time.
Module 2 · Optimization & the Vertex
Math 12-6 Vertex x-Coordinate of a Parabola
h=−b2ah = -\frac{b}{2a}
The x-coordinate of a parabola's vertex, sitting midway between the two roots and marking the axis…
Math 12-7 Discriminant of a Quadratic
Δ=b2−4ac\Delta = b^2 - 4ac
The quantity b² − 4ac under the square root, which reveals how many real roots a quadratic has…
Math 12-8 Vertex y-Coordinate of a Parabola
k=c−b24ak = c - \frac{b^2}{4a}
The y-coordinate of a parabola's vertex: its minimum value when a is positive, and its maximum when…
Math 12-9 Quadratic Formula (Positive Root)
x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}
The larger of the two solutions of ax² + bx + c = 0, taking the plus branch of the square root in…
Math 12-10 Quadratic Formula (Negative Root)
x=−b−b2−4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a}
The smaller of the two solutions of ax² + bx + c = 0, taking the minus branch of the square root in…
Math 12-11 Sum of the Roots of a Quadratic
S=−baS = -\frac{b}{a}
Vieta's relation giving the sum of a quadratic's two roots directly from its coefficients, without…
Math 12-12 Product of the Roots of a Quadratic
P=caP = \frac{c}{a}
Vieta's relation giving the product of a quadratic's two roots directly from its coefficients,…
Math 12-13 Completing the Square: the Constant Needed
k=(b2)2k = \left( \frac{b}{2} \right)^{2}
The constant you add to x² + bx to turn it into a perfect square trinomial, the pivotal step in…
Math 12-14 Break-Even Quantity
Q=Fp−vQ = \frac{F}{p - v}
Units that must be sold before fixed costs are covered: fixed cost divided by the contribution…
Module 3 · Vectors
Math 12-15 Magnitude of a 2D Vector
∣v⃗∣=vx2+vy2|\vec{v}| = \sqrt{v_x^2 + v_y^2}
Finds the length of a two-dimensional vector from its x- and y-components, the Pythagorean theorem…
Math 12-16 Direction Angle of a 2D Vector
θ=atan2⁡(vy,  vx)\theta = \operatorname{atan2}(v_y,\; v_x)
Gives the direction a two-dimensional vector points, measured counterclockwise from the positive…
Math 12-17 x-Component from Magnitude and Angle
vx=∣v⃗∣cos⁡θv_x = |\vec{v}| \cos\theta
Resolves a vector into its horizontal part from the vector's length and the angle it makes with the…
Math 12-18 y-Component from Magnitude and Angle
vy=∣v⃗∣sin⁡θv_y = |\vec{v}| \sin\theta
Resolves a vector into its vertical part from the vector's length and the angle it makes with the…
Math 12-19 Right-Triangle Tangent Ratio (TOA)
tan⁡θ=oa\tan\theta = \frac{o}{a}
Relates an acute angle of a right triangle to the ratio of its opposite and adjacent legs.
Math 12-20 Resultant of Two Vectors at an Angle
R=A2+B2+2ABcos⁡θR = \sqrt{A^2 + B^2 + 2AB\cos\theta}
Finds the magnitude of the sum of two vectors from their lengths and the angle between them, the…
Math 12-21 Law of Cosines
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
Finds the third side of any triangle from two sides and their included angle, or the angle from all…
Math 12-22 Dot Product of Two 2D Vectors (Components)
a⃗⋅b⃗=axbx+ayby\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y
Multiplies two plane vectors component by component and adds the results, giving the scalar that…
Math 12-23 Dot Product from Magnitudes and Included Angle
a⃗⋅b⃗=∣a⃗∣ ∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta
Gives the scalar product of two vectors from their lengths and the angle between them, the…
Math 12-24 Angle Between Two 2D Vectors (Components)
θ=arccos⁡(axbx+aybyax2+ay2 bx2+by2)\theta = \arccos\left(\frac{a_x b_x + a_y b_y}{\sqrt{a_x^2+a_y^2}\,\sqrt{b_x^2+b_y^2}}\right)
Finds the angle separating two plane vectors directly from their four components, by way of the…
Math 12-25 Cross Product Magnitude
∣a⃗×b⃗∣=∣a⃗∣ ∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta
Gives the length of the cross product of two vectors from their magnitudes and the angle between…
Math 12-26 Parallelogram Area from Two Vectors
A=∣axby−aybx∣A = \left|a_x b_y - a_y b_x\right|
Finds the area of the parallelogram spanned by two plane vectors directly from their four…
Math 12-27 Triangle Area from Two Vectors
A=12∣axby−aybx∣A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|
Finds the area of the triangle formed by two plane vectors from a shared vertex, using only their…
Math 12-28 Work from Force and Displacement Components
W=Fxdx+FydyW = F_x d_x + F_y d_y
Computes the work done by a force from the components of the force and the displacement, without…
Math 12-29 Scalar Projection of One Vector onto Another
compb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}
Measures how far a vector reaches along the direction of another, the length of the shadow it casts…
Module 4 · Counting & Probability
Math 12-30 Classical Probability
P=fnP = \frac{f}{n}
Probability of an event as the number of favourable outcomes divided by the total number of equally…
Math 12-31 Odds and Probability
O=P1−PO = \frac{P}{1 - P}
Converts between a probability and odds in favour, the ratio of the chance it happens to the chance…
Math 12-32 Complement Rule
P(Ac)=1−P(A)P(A^{c}) = 1 - P(A)
The chance an event does not happen is one minus the chance it does, because every trial must end…
Math 12-33 Addition Rule (Mutually Exclusive Events)
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
For events that cannot both happen, the chance that either one occurs is simply the sum of their…
Math 12-34 General Addition Rule
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
The chance that either of two events happens, correcting the simple sum by subtracting the overlap…
Math 12-35 Multiplication Rule (Independent Events)
P(A∩B)=P(A) P(B)P(A \cap B) = P(A) \, P(B)
When one event has no influence on the other, the chance that both occur is the product of their…
Math 12-36 Conditional Probability
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
The chance of A once B is known to have happened, found by rescaling the overlap to the reduced…
Math 12-37 General Multiplication Rule
P(A∩B)=P(A) P(B∣A)P(A \cap B) = P(A) \, P(B \mid A)
Chance that both events happen when the second depends on the first, as in drawing two cards…
Math 12-38 Factorial
n!=n×(n−1)×⋯×2×1n! = n \times (n-1) \times \cdots \times 2 \times 1
The product of every whole number from n down to 1, counting the ways that n distinct objects can…
Math 12-39 Permutations (nPr)
nPr=n!(n−r)!{}_{n}P_{r} = \frac{n!}{(n-r)!}
Number of ways to choose r items from n distinct items when the order of the selection matters, as…
Math 12-40 Combinations (nCr)
nCr=(nr)=n!r! (n−r)!{}_{n}C_{r} = \binom{n}{r} = \frac{n!}{r! \, (n-r)!}
Number of ways to choose r items from n distinct items when the order does not matter, as with…
Math 12-41 Binomial Probability
P(X=k)=(nk)pk(1−p) n−kP(X = k) = \binom{n}{k} p^{k} (1 - p)^{\,n-k}
Chance of exactly k successes in n independent trials that each succeed with the same fixed…
Math 12-42 Binomial Distribution Mean
μ=np\mu = n p
Expected number of successes across n independent trials that each succeed with probability p: the…
Math 12-43 Binomial Distribution Variance
σ2=np(1−p)\sigma^{2} = n p (1 - p)
Spread of the number of successes across n independent trials, at its largest when the per-trial…
Math 12-44 Probability of At Least One Success
P=1−(1−p)nP = 1 - (1 - p)^{n}
Chance that at least one of n independent attempts succeeds, found as one minus the chance that…
Math 12-45 Expected Value of a Bet
E=p W−(1−p) LE = p \, W - (1 - p) \, L
Average profit per play of a two-outcome wager that pays W with probability p and costs L the rest…
Math 12-46 Geometric Distribution (First Success)
P(X=k)=(1−p) k−1pP(X = k) = (1 - p)^{\,k-1} p
Chance that the first success in a run of repeated independent trials arrives exactly on trial…
Module 5 · Statistics & Data
Math 12-47 Z-Score (Standard Score)
z=x−μσz = \frac{x - \mu}{\sigma}
How many standard deviations a value sits above or below the mean, turning any measurement into a…
Math 12-48 Interquartile Range (IQR)
IQR=Q3−Q1IQR = Q_3 - Q_1
The width of the middle half of a data set, from the first quartile to the third, and the spread…
Math 12-49 Range (Max minus Min)
R=xmax⁡−xmin⁡R = x_{\max} - x_{\min}
The simplest measure of spread in a data set: the distance from the smallest observation to the…
Math 12-50 Variance and Standard Deviation
σ2=σ⋅σ\sigma^2 = \sigma \cdot \sigma
Converts between variance and standard deviation: the variance is the square of the standard…
Math 12-51 Outlier Lower Fence
LF=Q1−1.5 IQRLF = Q_1 - 1.5 \, IQR
Tukey's lower cutoff for outliers: any observation below one and a half interquartile ranges under…
Math 12-52 Outlier Upper Fence
UF=Q3+1.5 IQRUF = Q_3 + 1.5 \, IQR
Tukey's upper cutoff for outliers: any observation above one and a half interquartile ranges beyond…
Math 12-53 Regression Slope from Correlation
b=rsysxb = r \frac{s_y}{s_x}
The least-squares slope of a regression line, recovered from the correlation coefficient and the…
Math 12-54 Regression Line Intercept
a=yˉ−bxˉa = \bar{y} - b \bar{x}
The y-intercept of a least-squares line, fixed by the requirement that the line pass through the…
Math 12-55 Predicted Value from a Regression Line
y^=a+bx\hat{y} = a + b x
Reads a prediction off a fitted least-squares line for any chosen value of the predictor, given the…
Math 12-56 Coefficient of Determination (R²)
R2=r2R^{2} = r^{2}
The share of variation in y explained by a simple linear regression, obtained by squaring the…
Math 12-57 Percent Error
PE=∣xmeas−xacc∣xaccPE = \frac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}
How far a measurement strays from the accepted value, expressed as a fraction of that accepted…
Math 12-58 Weighted Mean of Two Groups
xˉ=n1xˉ1+n2xˉ2n1+n2\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}
Combines the averages of two groups into one overall mean, weighting each group by how many…
Module 6 · Financial Math
Math 12-59 Simple Interest
I=P r tI = P \, r \, t
Interest earned on a principal at a flat rate per period, without compounding.
Math 12-60 Percent Change
c=xnew−xoldxoldc = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}
Relative change from an old value to a new one, as a fraction of the old value.
Math 12-61 Compound Interest (Periodic)
A=P(1+rn)ntA = P \left( 1 + \frac{r}{n} \right)^{n t}
Amount after t periods when interest compounds n times per period at rate r.
Math 12-62 Exponential Growth
A=A0(1+r)tA = A_0 (1 + r)^{t}
Amount after t periods of compound growth at rate r per period.
Math 12-63 Present Value
PV=FV(1+r)t\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}
Today's value of a future amount, discounted at rate r per period.
Math 12-64 Rule of 72 (Doubling Time)
n≈0.72in \approx \frac{0.72}{i}
Mental-arithmetic estimate of how many periods it takes money to double at rate i: with the rate…
Math 12-65 Exponential Growth by Doubling Time
N=N0⋅2t/TN = N_0 \cdot 2^{t/T}
Growth of a quantity that doubles every fixed interval T.
Math 12-66 Solve an Exponential Equation for the Exponent
x=ln⁡(y/a)ln⁡bx = \frac{\ln (y / a)}{\ln b}
Solves a·bˣ = y for the unknown exponent x using logarithms, the standard move behind every growth…
Math 12-67 Future Value of an Annuity (Regular Deposits)
FV=D (1+i)n−1i\mathit{FV} = D\,\frac{(1+i)^n - 1}{i}
What an equal deposit made at the end of every period grows to after n periods at rate i: the…
Math 12-68 Loan Payment (Amortized Loan or Mortgage)
M=P i1−(1+i)−nM = \frac{P\,i}{1 - (1+i)^{-n}}
Level payment that retires a loan of principal P in exactly n payments at periodic interest rate i:…
Math 12-69 Total Interest Paid Over a Loan
I=M n−PI = M\,n - P
Everything a loan costs beyond the amount borrowed: total of all payments minus the principal.
Math 12-70 Effective Annual Rate from a Nominal Rate
EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1
What a quoted nominal annual rate really costs or earns once it compounds m times a year: 12%…