Parabola standard forms
$$(x-h)^2=4p(y-k),\qquad(y-k)^2=4p(x-h)$$
Describe a parabola by vertex and focal distance.
High school · APPrecalculus formulas, grouped by topic.
10 precalculus formulas across 5 topics, free and without an account. Precalculus formulas prepare the analytic groundwork for calculus. Conic sections are described by their centers, axes, and focal distances; complex numbers gain a modulus and a polar form; and infinite geometric series introduce the idea of convergence. Each of these is a statement about a function's shape, which is why sketching before substituting usually reveals whether an answer is plausible.
Every precalculus formula on one page
Each entry links to its own page, where the symbols are defined, the conditions are stated, and the formula is substituted through a worked example. Levels covered here: High school, AP, University.
Conics
3 formulas in conics.
Ellipse standard form
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$
Describe an axis-aligned ellipse centered at $(h,k)$.
High school · APHyperbola standard form
$$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$
Describe a horizontal axis-aligned hyperbola.
High school · APSeries
1 formula in series.
Infinite geometric series
$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}$$
Add infinitely many terms of a convergent geometric sequence.
High school · APComplex numbers
3 formulas in complex numbers.
Complex modulus
$$|a+bi|=\sqrt{a^2+b^2}$$
Measure a complex number’s distance from the origin.
High school · APComplex polar form
$$z=r(\cos\theta+i\sin\theta)=re^{i\theta},\qquad r=\sqrt{x^2+y^2}$$
Represent a complex number by magnitude and direction.
High school · UniversityDe Moivre’s theorem
$$[r(\cos\theta+i\sin\theta)]^n=r^n[\cos(n\theta)+i\sin(n\theta)]$$
Raise a complex number in polar form to an integer power.
High school · UniversityFunctions
2 formulas in functions.
Function composition
$$(f\circ g)(x)=f(g(x))$$
Use the output of one function as the input of another.
High schoolInverse function relationship
$$f^{-1}(f(x))=x,\qquad f(f^{-1}(x))=x$$
Undo a one-to-one function with its inverse.
High schoolTransformations
1 formula in transformations.
Function transformation
$$y=a\,f(b(x-h))+k$$
Describe vertical scale, horizontal scale, reflection, and translation of a graph.
High schoolKeep going with precalculus
A formula sheet is a reference, not a method. Use the subject guide to see where these relationships come from, a calculator to check a substitution you are unsure about, and practice questions to find out whether you can actually retrieve them.
- Study precalculus — the subject hub with lessons, worked methods, and common mistakes.
- Precalculus calculator — enter your own values and compare the result with your working.
- Precalculus Practice — retrieval practice across every topic listed above.
- Math formula library — search all 215 formulas by name, symbol, or condition.
Formulas in the other subjects
Mathematics does not stop at a subject boundary, and neither do its formulas. Each subject page below lists its own relationships, conditions, and worked examples.
Arithmetic
Review the core arithmetic relationships, their restrictions, and worked substitutions.
Algebra
Review the core algebra relationships, their restrictions, and worked substitutions.
Geometry
Review the core geometry relationships, their restrictions, and worked substitutions.
Trigonometry
Review the core trigonometry relationships, their restrictions, and worked substitutions.
Calculus
Review the core calculus relationships, their restrictions, and worked substitutions.
Statistics
Review the core statistics relationships, their restrictions, and worked substitutions.
Probability
Review the core probability relationships, their restrictions, and worked substitutions.
Linear Algebra
Review the core linear algebra relationships, their restrictions, and worked substitutions.
Differential Equations
Review the core differential equations relationships, their restrictions, and worked substitutions.
Discrete Math
Review the core discrete math relationships, their restrictions, and worked substitutions.