Browse all topics
Search by keyword, filter by level or learned status, or pick a subject. Topics are grouped by level, starting with beginner.
1000 topics
🌱 Beginner Core ideas and first intuitions. No prior knowledge needed.
Linear Functions
A straight line with constant rate of change. The slope m tilts it and the intercept b shifts it vertically.
Slope as Rate of Change
Slope measures how much y changes per unit of x — steeper lines mean faster change.
Absolute Value Functions
Distance from zero creates a V shape. Shifts and stretches move and narrow the V.
Quadratic Functions
Parabolas model projectiles and optimisation. The sign of a decides whether it opens up or down.
Vertex Form of a Parabola
Writing a quadratic as a(x−h)²+k reveals its turning point (h, k) at a glance.
Roots and the Discriminant
The discriminant b²−4ac tells you whether a quadratic crosses the x-axis twice, once, or never.
Cubic Polynomials
Cubics can have up to two turning points and always cross the x-axis at least once.
End Behaviour of Polynomials
For large |x| the highest power dominates. Odd degrees go opposite ways, even degrees go the same way.
Rational Functions
A ratio of polynomials can have vertical asymptotes where the denominator vanishes.
Square Root Functions
The inverse of a half-parabola: slow, steady growth that starts at a point.
Exponential Growth
Quantities that multiply by a constant factor each step: money, bacteria, viral spread.
Exponential Decay
The opposite of growth: radioactive material, drug concentration and cooling follow e^(−kt).
Logarithms
A logarithm answers “what power gives this number?” It is the inverse of an exponential.
Euler’s Number e
The base for which the slope of eˣ equals its height, making it the natural language of growth.
Inverse Functions
Swapping inputs and outputs reflects a function across the line y = x.
Piecewise Functions
Different rules on different intervals, like tax brackets or shipping rates.
Floor and Step Functions
Rounding down produces staircases — the model for pricing tiers and digital sampling.
Even and Odd Functions
Even graphs mirror across the y-axis; odd graphs are unchanged by a half-turn about the origin.
Composition of Functions
Feed one function’s output into another. Order matters: f(g(x)) ≠ g(f(x)).
Systems of Linear Equations
Where two lines cross is the one point that satisfies both equations at once.
Logistic Growth
Growth that levels off at a carrying capacity — an S-curve found in populations and technology adoption.
Inverse Proportion
When one quantity doubles the other halves; the product stays constant.
Direct and Power Variation
y ∝ xⁿ describes areas (n=2), volumes (n=3) and many scaling laws in nature.
Arithmetic Sequences
Add the same amount each step: the terms lie on a straight line.
Geometric Sequences
Multiply by the same ratio each step; terms grow or shrink exponentially.
Triangular Numbers
Counting dots in growing triangles gives 1, 3, 6, 10, … with a neat closed form.
Recurrence Relations
Each term is defined from the previous: the discrete cousin of differential equations.
Circles
All points a fixed distance from a centre. Parametric equations trace the circle as an angle sweeps.
Ellipses
A stretched circle: the sum of distances to two foci stays constant. Planets travel on ellipses.
Parabolas as Conics
Slicing a cone parallel to its side gives a parabola — the shape of satellite dishes.
Hyperbolas
Two mirror-image branches; the difference of distances to the foci is constant.
Lissajous Figures
Two perpendicular oscillations with related frequencies draw looping figures on oscilloscopes.
Archimedean Spiral
Radius grows steadily with angle, as on a coiled rope or a vinyl groove.
Polar Roses
Petal curves from r = cos(kθ). Odd k gives k petals, even k gives 2k.
Cardioid
A heart-shaped curve traced by a point on one circle rolling around another of equal size.
Lemniscate of Bernoulli
The figure-eight curve that inspired the infinity symbol.
Catenary
A hanging chain forms a cosh curve, not a parabola. Arches use the same shape upside down.
Lissajous Knots
Add a phase offset and the figure appears to rotate in three dimensions.
Unit Circle
Every angle corresponds to a point (cos θ, sin θ) on the unit circle.
The Tangent Function
Tangent is the slope of the radius, shooting to infinity when cosine is zero.
Radians
An angle measured by arc length on the unit circle: 2π radians make a full turn.
Amplitude, Period and Phase
Every sinusoid is described by height, length of one cycle and horizontal shift.
Double-Angle Formulas
Doubling an angle is equivalent to a product of sine and cosine of the original.
Mean Value Theorem
Somewhere between two points the tangent slope equals the secant slope.
Area Under a Parabola
Archimedes found the area under a parabola; calculus does it in one line.
Area Between Curves
Subtract the lower function from the upper one and integrate the gap.
Determinants and Area
The determinant is the factor by which a matrix scales area; negative means a flip.
Slope–Intercept from Two Points
Two points fix a line: the slope is rise over run and the intercept follows.
Point–Slope Form
A line through a known point (h, k) with slope m is easiest written around that point.
Parallel and Perpendicular Lines
Parallel lines share a slope; perpendicular slopes are negative reciprocals.
Linear Inequalities
A boundary line splits the plane; the solution is the shaded half on one side.
Completing the Square
Rewrite x² + bx as a perfect square plus a constant to find the vertex.
The Quadratic Formula
The roots of any quadratic come from one formula; the axis of symmetry sits halfway between them.
Factoring and Zeros
A polynomial crosses the axis where each factor is zero.
Polynomial Multiplicity
A repeated root makes the graph touch the axis and bounce instead of crossing.
Rational Root Intuition
Dividing by a linear factor leaves a hole or an asymptote at its zero.
Horizontal Asymptotes
A rational function levels off at a ratio of leading coefficients.
Slant Asymptotes
When the numerator is one degree higher, the graph follows a slanted line.
Cube Root and Odd Roots
Odd roots are defined for negatives and pass through the origin.
Exponent Rules in Pictures
Multiplying powers adds exponents: the graphs of xᵃ·xᵇ and xᵃ⁺ᵇ coincide.
Frequently asked questions
How do I search?
Type one or more words. Every word must match a topic’s title, summary, formula or subject, so “wave sound” finds topics that mention both.
What do the level filters mean?
Beginner: Core ideas and first intuitions. No prior knowledge needed. Intermediate: Builds on the basics with relationships, methods and trade-offs. Advanced: Deeper models, formal results and harder reasoning.
How do I see only the topics I have not learned?
Choose “To learn” in the status filter. Choose “Learned” to see the ones you have finished.