Formula Explorer

Every formula. Hover every variable.

Search the entire sheet. Hover any variable for its meaning, role, and units.

Quadratic Vertex Form

quadratics
y=a(x−h)2+ky = a(x-h)^2 + k

Used to easily find the vertex and axis of symmetry.

Quadratic Formula

quadratics
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

Solves any quadratic equation $ax^2 + bx + c = 0$ for exact roots.

Discriminant

quadratics
Δ=b2−4ac\Delta = b^2 - 4ac

Determines the number of real roots without solving the equation.

Multiplying Radicals

absolute value
(ab)(cd)=acbd(a\sqrt{b})(c\sqrt{d}) = ac\sqrt{bd}

Multiply coefficients together and radicands together.

General Transformation

transformations
y=a f(k(x−d))+cy = a\,f\big(k(x-d)\big) + c

Applies vertical/horizontal stretches, reflections, and translations to a parent function.

Dividing Rationals

rationals
AB÷CD=A⋅DB⋅C\frac{A}{B} \div \frac{C}{D} = \frac{A \cdot D}{B \cdot C}

Keep the first fraction, change to multiplication, flip the second fraction.

Compound Interest

exponentials
A=P(1+rn)ntA = P\left(1+\frac{r}{n}\right)^{nt}

Calculates interest earned on both the principal and accumulated interest.

Half-Life Formula

exponentials
A=a(12)thA = a\left(\frac{1}{2}\right)^{\frac{t}{h}}

Models radioactive decay where mass halves every $h$ units of time.

Sine Law

trigonometry
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Used on non-right triangles. Requires an angle and its opposite side.

Cosine Law

trigonometry
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Used on non-right triangles. Requires two sides and the contained angle, or 3 sides.

Sinusoidal Function

trigonometry
y=asin⁡(k(x−d))+cy = a \sin(k(x-d)) + c

Models periodic behavior like tides or Ferris wheels.

Arithmetic nth Term

sequences
tn=a+(n−1)dt_n = a + (n-1)d

Finds any term in an arithmetic sequence.

Geometric nth Term

sequences
tn=a⋅rn−1t_n = a \cdot r^{n-1}

Finds any term in a geometric sequence.

Arithmetic Series Sum

sequences
Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)

Sums the first $n$ terms of an arithmetic sequence.

Geometric Series Sum

sequences
Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}

Sums the first $n$ terms of a geometric sequence.