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Grade 11 Math: Functions & Applications: formula sheet

Quadratics, growth and decay, periodic motion, trigonometry and money · 43 formulas · metric edition 1

Download the PDF: 2 pagesThe full study guide: modules, problems, answer key
Module 1 · Quadratic Functions
Math: Functions & Applications 11-1 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: Functions & Applications 11-2 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: Functions & Applications 11-3 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: Functions & Applications 11-4 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: Functions & Applications 11-5 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: Functions & Applications 11-6 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: Functions & Applications 11-7 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: Functions & Applications 11-8 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: Functions & Applications 11-9 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: Functions & Applications 11-10 Projectile Maximum Height
H=v02sin⁡2θ2gH = \frac{v_0^{2} \sin^{2}\theta}{2g}
Peak height reached by a projectile launched at a given speed and angle above level ground,…
Math: Functions & Applications 11-11 Projectile Time of Flight
T=2v0sin⁡θgT = \frac{2 v_0 \sin\theta}{g}
Total time a projectile stays airborne before returning to its launch height, set by launch speed…
Module 2 · Exponential Growth & Decay
Math: Functions & Applications 11-12 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: Functions & Applications 11-13 Exponential Decay
A=A0(1−r)tA = A_0 (1 - r)^{t}
Amount remaining after t periods of losing a fixed fraction r per period.
Math: Functions & Applications 11-14 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: Functions & Applications 11-15 Declining-Balance Depreciation (Book Value)
B=C (1−d)kB = C\,(1 - d)^k
Book value of an asset after k years when a fixed fraction d of the remaining value is written off…
Math: Functions & Applications 11-16 Half-Life Decay
N=N0(12)t/t1/2N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}
Gives the quantity remaining after repeated halvings over an elapsed time measured in half-lives.
Math: Functions & Applications 11-17 Half-Life and Decay Constant
t1/2=ln⁡2λt_{1/2} = \frac{\ln 2}{\lambda}
Converts between a half-life and the exponential decay constant via the factor ln 2 = 0.6931471806.
Math: Functions & Applications 11-18 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: Functions & Applications 11-19 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: Functions & Applications 11-20 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: Functions & Applications 11-21 Logarithm Change of Base
log⁡bx=ln⁡xln⁡b\log_b x = \frac{\ln x}{\ln b}
Value of a logarithm in any base, rewritten with natural logarithms so an ordinary calculator can…
Module 3 · Periodic Functions
Math: Functions & Applications 11-22 Period-Frequency Relation
T=1fT = \frac{1}{f}
Period and frequency are reciprocals: seconds per cycle versus cycles per second.
Math: Functions & Applications 11-23 Wave Speed (v = fλ)
v=fλv = f \lambda
The universal wave equation: a wave's speed equals its frequency times its wavelength.
Math: Functions & Applications 11-24 Simple Pendulum Period
T=2πLgT = 2\pi \sqrt{\frac{L}{g}}
Period of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).
Math: Functions & Applications 11-25 Speed in Circular Motion (v = 2πr/T)
v=2πrTv = \frac{2\pi r}{T}
Speed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per…
Math: Functions & Applications 11-26 SHM Displacement at Time t
x=Acos⁡(ωt)x = A \cos\left(\omega t\right)
Displacement of a simple harmonic oscillator at time t, a cosine of amplitude A and angular…
Math: Functions & Applications 11-27 Period of a Spring-Mass Oscillator
T=2πmkT = 2\pi \sqrt{\tfrac{m}{k}}
Period of a mass bouncing on a spring, set only by the mass and the spring stiffness.
Module 4 · Trigonometry Applications
Math: Functions & Applications 11-28 Right-Triangle Sine Ratio (SOH)
sin⁡θ=oh\sin\theta = \frac{o}{h}
Relates an acute angle of a right triangle to its opposite side and the hypotenuse.
Math: Functions & Applications 11-29 Right-Triangle Cosine Ratio (CAH)
cos⁡θ=ah\cos\theta = \frac{a}{h}
Relates an acute angle of a right triangle to its adjacent side and the hypotenuse.
Math: Functions & Applications 11-30 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: Functions & Applications 11-31 Pythagorean Theorem
a2+b2=c2a^{2} + b^{2} = c^{2}
Relates the three sides of a right triangle.
Math: Functions & Applications 11-32 Law of Sines
asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}
In any triangle, each side is proportional to the sine of its opposite angle.
Math: Functions & Applications 11-33 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: Functions & Applications 11-34 Triangle Area (Two Sides and Included Angle)
A=12 absin⁡CA = \tfrac{1}{2}\,ab\sin C
Computes a triangle's area from two sides and the angle between them, with no height needed.
Module 5 · Sequences & Financial Math
Math: Functions & Applications 11-35 Arithmetic Sequence nth Term
an=a1+(n−1)da_n = a_1 + (n - 1) d
The nth term of a sequence that steps by a fixed common difference.
Math: Functions & Applications 11-36 Geometric Sequence nth Term
an=a1 r n−1a_n = a_1 \, r^{\,n-1}
The nth term of a sequence that multiplies by a fixed ratio each step.
Math: Functions & Applications 11-37 Arithmetic Series Sum
Sn=n2(2a1+(n−1)d)S_n = \frac{n}{2} \left( 2 a_1 + (n - 1) d \right)
Sum of the first n terms of an arithmetic sequence.
Math: Functions & Applications 11-38 Geometric Series Sum
Sn=a1(1−rn)1−rS_n = \frac{a_1 (1 - r^{n})}{1 - r}
Sum of the first n terms of a geometric sequence.
Math: Functions & Applications 11-39 Simple Interest
I=P r tI = P \, r \, t
Interest earned on a principal at a flat rate per period, without compounding.
Math: Functions & Applications 11-40 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: Functions & Applications 11-41 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: Functions & Applications 11-42 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: Functions & Applications 11-43 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:…