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Grade 12 Math — formula sheet

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

The full booklet — learning zone, problems, answer key
Slope Between Two Points
m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
Displacement from Average Velocity
d=v0+v2 td = \frac{v_0 + v}{2} \, t
Final Velocity (Uniform Acceleration)
v=v0+atv = v_0 + a t
Displacement (Uniform Acceleration)
d=v0t+12at2d = v_0 t + \tfrac{1}{2} a t^2
Velocity-Displacement Relation (v² = v₀² + 2ad)
v2=v02+2adv^2 = v_0^2 + 2 a d
Vertex x-Coordinate of a Parabola
h=−b2ah = -\frac{b}{2a}
Discriminant of a Quadratic
Δ=b2−4ac\Delta = b^2 - 4ac
Vertex y-Coordinate of a Parabola
k=c−b24ak = c - \frac{b^2}{4a}
Quadratic Formula (Positive Root)
x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}
Quadratic Formula (Negative Root)
x=−b−b2−4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a}
Sum of the Roots of a Quadratic
S=−baS = -\frac{b}{a}
Product of the Roots of a Quadratic
P=caP = \frac{c}{a}
Completing the Square: the Constant Needed
k=(b2)2k = \left( \frac{b}{2} \right)^{2}
Break-Even Quantity
Q=Fp−vQ = \frac{F}{p - v}
Magnitude of a 2D Vector
∣v⃗∣=vx2+vy2|\vec{v}| = \sqrt{v_x^2 + v_y^2}
Direction Angle of a 2D Vector
θ=atan2⁡(vy,  vx)\theta = \operatorname{atan2}(v_y,\; v_x)
x-Component from Magnitude and Angle
vx=∣v⃗∣cos⁡θv_x = |\vec{v}| \cos\theta
y-Component from Magnitude and Angle
vy=∣v⃗∣sin⁡θv_y = |\vec{v}| \sin\theta
Right-Triangle Tangent Ratio (TOA)
tan⁡θ=oa\tan\theta = \frac{o}{a}
Resultant of Two Vectors at an Angle
R=A2+B2+2ABcos⁡θR = \sqrt{A^2 + B^2 + 2AB\cos\theta}
Law of Cosines
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
Dot Product of Two 2D Vectors (Components)
a⃗⋅b⃗=axbx+ayby\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y
Dot Product from Magnitudes and Included Angle
a⃗⋅b⃗=∣a⃗∣ ∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta
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)
Cross Product Magnitude
∣a⃗×b⃗∣=∣a⃗∣ ∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta
Parallelogram Area from Two Vectors
A=∣axby−aybx∣A = \left|a_x b_y - a_y b_x\right|
Triangle Area from Two Vectors
A=12∣axby−aybx∣A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|
Work from Force and Displacement Components
W=Fxdx+FydyW = F_x d_x + F_y d_y
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}|}
Classical Probability
P=fnP = \frac{f}{n}
Odds and Probability
O=P1−PO = \frac{P}{1 - P}
Complement Rule
P(Ac)=1−P(A)P(A^{c}) = 1 - P(A)
Addition Rule (Mutually Exclusive Events)
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)
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)
Multiplication Rule (Independent Events)
P(A∩B)=P(A) P(B)P(A \cap B) = P(A) \, P(B)
Conditional Probability
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
General Multiplication Rule
P(A∩B)=P(A) P(B∣A)P(A \cap B) = P(A) \, P(B \mid A)
Factorial
n!=n×(n−1)×⋯×2×1n! = n \times (n-1) \times \cdots \times 2 \times 1
Permutations (nPr)
nPr=n!(n−r)!{}_{n}P_{r} = \frac{n!}{(n-r)!}
Combinations (nCr)
nCr=(nr)=n!r! (n−r)!{}_{n}C_{r} = \binom{n}{r} = \frac{n!}{r! \, (n-r)!}
Binomial Probability
P(X=k)=(nk)pk(1−p) n−kP(X = k) = \binom{n}{k} p^{k} (1 - p)^{\,n-k}
Binomial Distribution Mean
μ=np\mu = n p
Binomial Distribution Variance
σ2=np(1−p)\sigma^{2} = n p (1 - p)
Probability of At Least One Success
P=1−(1−p)nP = 1 - (1 - p)^{n}
Expected Value of a Bet
E=p W−(1−p) LE = p \, W - (1 - p) \, L
Geometric Distribution (First Success)
P(X=k)=(1−p) k−1pP(X = k) = (1 - p)^{\,k-1} p
Z-Score (Standard Score)
z=x−μσz = \frac{x - \mu}{\sigma}
Interquartile Range (IQR)
IQR=Q3−Q1IQR = Q_3 - Q_1
Range (Max minus Min)
R=xmax⁡−xmin⁡R = x_{\max} - x_{\min}
Variance and Standard Deviation
σ2=σ⋅σ\sigma^2 = \sigma \cdot \sigma
Outlier Lower Fence
LF=Q1−1.5 IQRLF = Q_1 - 1.5 \, IQR
Outlier Upper Fence
UF=Q3+1.5 IQRUF = Q_3 + 1.5 \, IQR
Regression Slope from Correlation
b=rsysxb = r \frac{s_y}{s_x}
Regression Line Intercept
a=yˉ−bxˉa = \bar{y} - b \bar{x}
Predicted Value from a Regression Line
y^=a+bx\hat{y} = a + b x
Coefficient of Determination (R²)
R2=r2R^{2} = r^{2}
Percent Error
PE=∣xmeas−xacc∣xaccPE = \frac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}
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}
Simple Interest
I=P r tI = P \, r \, t
Percent Change
c=xnew−xoldxoldc = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}
Compound Interest (Periodic)
A=P(1+rn)ntA = P \left( 1 + \frac{r}{n} \right)^{n t}
Exponential Growth
A=A0(1+r)tA = A_0 (1 + r)^{t}
Present Value
PV=FV(1+r)t\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}
Rule of 72 (Doubling Time)
n≈0.72in \approx \frac{0.72}{i}
Exponential Growth by Doubling Time
N=N0⋅2t/TN = N_0 \cdot 2^{t/T}
Solve an Exponential Equation for the Exponent
x=ln⁡(y/a)ln⁡bx = \frac{\ln (y / a)}{\ln b}
Future Value of an Annuity (Regular Deposits)
FV=D (1+i)n−1i\mathit{FV} = D\,\frac{(1+i)^n - 1}{i}
Loan Payment (Amortized Loan or Mortgage)
M=P i1−(1+i)−nM = \frac{P\,i}{1 - (1+i)^{-n}}
Total Interest Paid Over a Loan
I=M n−PI = M\,n - P
Effective Annual Rate from a Nominal Rate
EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1