Interactive math labs: functions, calculus, conics, geometry and probability
Functions, calculus, conic sections, geometry and probability, each as a board you can drag.
4 suites · 20 chapters
Functions & calculus
Drag the parameters of a graph and watch it move: a parabola shifted and stretched, exponentials mirrored into logarithms, sine unrolled from the unit circle and reshaped by A, ω and φ, tangents that reveal where a function rises and turns, a zero trapped by halving, and rectangles adding up to an integral.
Vertex form: how a, h and k move a parabola
What do a, h and k each do to y = a(x − h)² + k?
Vertex form · Translations · Axis of symmetry
Open laby = aˣ and y = logₐ x: inverse functions mirrored in y = x
How are y = aˣ and y = logₐ x related, and what does the base a decide?
Exponential function · Logarithmic function · Inverse functions
Open labSine and cosine on the unit circle
Why does a point going round a circle draw a sine wave?
Unit circle · Radian measure · Special angles
Open laby = A sin(ωx + φ): amplitude, period and phase shift
What do A, ω and φ each do to the graph of y = sin x?
Amplitude · Period T = 2π/ω · Phase shift
Open labDerivative as the slope of a tangent
Why is the derivative the slope of the tangent line?
Secant slope · Limit definition · Tangent line
Open labMonotonicity and extrema: reading a function off the sign of f′
How does the sign of f′(x) tell you where f rises, where it falls, and where it turns?
Sign of the derivative · Intervals of increase · Local maximum and minimum
Open labZeros and the bisection method: trapping a root by halving
How can you find a zero of f(x) = x³ − 3x + 1 to two decimal places without solving the equation?
Zeros of a function · Intermediate value theorem · Bisection method
Open labIntegral as area: Riemann sums
How do thin rectangles add up to the exact area under a curve?
Riemann sum · Definite integral · Limit of a sum
Open labConic sections
Each conic from its own definition: a string of fixed length round two pins, equal distances to a focus and a line, a fixed difference of distances. Then one eccentricity ties the three into a single family.
The ellipse: a constant sum of distances to two foci
Why does a loop of string around two pins draw an ellipse, and where does b² = a² − c² come from?
Definition by two foci · Standard equation · b² = a² − c²
Open labThe parabola: equal distances to a focus and a directrix
Why is the focal radius of y² = 2px just x + p/2, and how long is a chord through the focus?
Focus and directrix · Standard equation y² = 2px · Focal radius
Open labThe hyperbola: a constant difference of distances, and its asymptotes
What stays constant on a hyperbola, and why do its branches hug the lines y = ±(b/a)x without touching them?
Definition by two foci · Standard equation · c² = a² + b²
Open labConic sections and eccentricity
How can one number turn a circle into an ellipse, a parabola and a hyperbola?
Focus and directrix · Eccentricity · Ellipse, parabola, hyperbola
Open labGeometry & proofs
Proofs you can take apart by hand: move the pieces and see why the theorem has to hold.
Pythagorean theorem: proof by rearrangement
Why is a² + b² = c² for every right triangle?
Right triangle · Area conservation · Proof without words
Open labInscribed angle theorem: half the arc it stands on
Why does every angle drawn from the circle onto the same arc come out the same, and exactly half the central angle?
Inscribed angle · Central angle · Angle in a semicircle
Open labSimilar triangles and homothety: lengths times k, area times k²
When a triangle is scaled from a point, why do its angles stay put, and why does the area grow by k² rather than k?
Homothety (dilation) · Similar triangles · Ratio of similitude
Open labVector addition and the dot product: from |a||b|cos θ to a₁b₁ + a₂b₂
The dot product is defined with lengths and an angle. Why does it come out as a₁b₁ + a₂b₂ in coordinates, where no angle appears?
Parallelogram rule · Dot product · Projection
Open labProbability & statistics
Chance built up from counting: choose and arrange, add up the paths of repeated trials, read probability as area under the bell curve, and watch a frequency settle as the trials pile up.
Permutations and combinations: count the orders, then divide them out
Why is the number of ways to choose k things from n equal to n(n−1)…(n−k+1) divided by k!, and why does Pascal’s triangle give the same numbers by adding?
Multiplication principle · Permutations · Combinations
Open labThe binomial distribution: count the paths, multiply the chances
Where does P(X = k) = C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ come from, and why is the mean np?
Independent trials · Binomial probability · Mean np
Open labThe normal distribution: 68%, 95% and 99.7% measured as areas
Why does the share within one, two or three standard deviations come out the same for every normal curve, whatever μ and σ are?
Normal density · Probability as area · 68–95–99.7 rule
Open labThe law of large numbers: why the frequency settles at p
A coin has no memory, so why does the share of heads close in on one half as the tosses pile up?
Frequency and probability · Variance of the mean p(1 − p)/n · Chebyshev's inequality
Open lab