Mathematics · functions
Functions and growth from a straight line to an explosion
A function is a recipe: put in x, get out y. A straight line describes a taxi fare, a parabola — the flight of a ball and a satellite dish, and an exponential function — bacteria, savings accounts and rumors. Drag points, shift graphs by a vector, race linear growth against exponential, multiply with a slide rule and find out which formula really fits the data.
Laboratorium · linear function
Equation of the line
Second line through point Q
Lines in everyday life
Guess the line
round 1 · total 0 ptsRead off the picture where the line crosses the y-axis (that is b) and how much it rises when you move 1 to the right (that is a).
Laboratorium · quadratic function
Parabolas in the world
Laboratorium · shifts and reflections of a graph
Base function f
Cheat sheet: what each transformation does
- f(x − p)shift by p to the right (when p > 0) — despite the minus sign!
- f(x) + qshift by q up; together with the previous one, a shift by the vector [p, q]
- a · f(x)vertical stretch by a factor of a (for a < 1 — flattening); the zeros stay put
- f(k · x)horizontal squeeze by a factor of k (for k < 1 — stretching)
- −f(x)reflection in the x-axis: the graph turns upside down
- f(−x)reflection in the y-axis: left and right swap places
- |f(x)|the parts below the x-axis flip above it
- f(|x|)the right half of the graph is mirrored onto the left side
Laboratorium · exponential growth and decay
Contestants
Legends of doubling
Compound interest
This is a model, not financial advice.Compounding of interest
| 100% a year, compounded | after a year from 1 PLN |
|---|
Exponential decay
Laboratorium · the logarithm, or “how many times do I multiply?”
Base of the logarithm
Guess the exponent
How many times do you have to multiply 2 by itself to get 1024? That is exactly log₂ 1024.
Slide rule
Linear or logarithmic axis?
Where logarithms live
Benford's law: why are there the most ones?
Laboratorium · which formula fits the data?
Data
Model
1 = perfect, 0 = explains nothing—
A short history of functions
- 1614John Napier publishes tables of logarithms — multiplication turns into addition. Astronomers could now calculate many times faster.
- about 1620–1624Henry Briggs proposes base-10 logarithms and publishes large tables of them; at about the same time Edmund Gunter draws a logarithmic scale, and William Oughtred makes a slide rule out of two such scales.
- 1637René Descartes joins algebra and geometry in “La Géométrie”: a curve can be described by an equation in a coordinate system.
- 1683Jakob Bernoulli studies compound interest with ever more frequent compounding — the limit it approaches is the number we now call e.
- 18th centuryLeonhard Euler popularized the notation f(x) and the symbol e, and described the exponential function and the logarithm the way we teach them today.
- Lwów, 1920–1939Stefan Banach and the Lwów School of Mathematics help create functional analysis — a field in which whole functions are treated like points in a space.
MathematicsHow to read a function
Function, domain, range
A function assigns to every argument x from its domain exactly one value y = f(x). All the values it produces form its range, and the points (x, f(x)) form its graph. A zero is an argument for which f(x) = 0: there the graph crosses the x-axis. A function is increasing when a larger argument gives a larger value, and decreasing when it is the other way round — that is its monotonicity.
Linear function
The slope a tells you how much y grows when x grows by 1, and the intercept b is where the line crosses the y-axis. Lines are parallel when they have equal a, and perpendicular when a1 · a2 = −1. A taxi fare, a phone plan and converting degrees Celsius to Fahrenheit (F = 1.8 · C + 32) are all linear functions: a fixed fee plus a fixed rate for every unit.
Quadratic function
These are three ways of writing the same parabola: standard, vertex (you can see the vertex W = (p, q) right away) and factored (you can see the zeros; it exists only when Δ ≥ 0). A parabola is symmetric about the line x = p. For a > 0 the arms point up, and the range is [q, +∞).
Shifting by a vector, and reflections
Subtracting p inside the parentheses shifts the graph to the right, because the value the function used to have at 0 it now reaches only at p. A minus in front of the function reflects the graph in the x-axis, a minus on the argument — in the y-axis.
Exponential growth and the number e
Linear growth adds the same amount each step; exponential growth multiplies by the same factor. That is why exponential growth always overtakes linear in the end — even when it starts much lower. When interest is added more and more often, (1 + 1/n)n approaches e ≈ 2.71828. The same formula works in reverse as decay: N = N0 · (½)t/T, where T is the half-life.
Logarithm
A logarithm answers the question “to what power must I raise the base?”. The logarithmic function is the inverse of the exponential one, so their graphs are symmetric about the line y = x. Because log(xy) = log x + log y, a slide rule multiplies by adding lengths. On a logarithmic scale every step of the same length means multiplying by the same number — that is how decibels, pH, and the magnitudes of stars and earthquakes work.
Simplifications on this page
Taxi fares and phone plans are examples. The ball's flight and the fountain ignore air resistance. Compound interest ignores capital gains tax (19% in Poland) and assumes a constant interest rate and constant inflation. The cooling coffee follows a simplified Newton's law of cooling, and the drug in the blood — the simplest one-compartment model with doses given instantly. The data in the “Functions in the world” mode are illustrative but have a realistic shape; day length in Warsaw is calculated from an astronomical formula, and the transistor counts come from manufacturers' data (rounded).
Did you know…?
- If you could fold a sheet of paper 0.1 mm thick in half 42 times, it would be 0.1 mm · 2⁴² ≈ 440,000 km thick — more than the average distance to the Moon (about 384,400 km). In practice you can't fold a sheet more than a dozen or so times.
- Apollo program engineers did their everyday calculations on slide rules, and the astronauts carried a small slide rule on board as a backup “calculator”.
- In many real data sets about 30% of the numbers start with 1, and fewer than 5% with 9 (Benford's law). Auditors use this to catch made-up numbers in accounts: people invent first digits too evenly.
Challenges
- A line through both targetsIn “Line” mode, set a line that passes through both gold stars.not completed
- Parabola to orderSet a parabola with its vertex at (2, −3) and one zero at 5. Where does the other one land?not completed
- Bull's-eyeSolve the puzzle in transformation mode: match your graph to the dashed target.not completed
- The overtaking momentSet the step n at which exponential growth first overtakes linear.not completed
- Multiplying without a calculatorDo a task on the slide rule: move scale C and the cursor so you can read off the product.not completed
- The right modelChoose a model that makes sense for the data and get a fit of R² > 0.95.not completed
Challenges check themselves. Progress is saved in this browser.
See also
A derivative is the slope of a tangent, sine is a periodic function, and radioactive decay, chemical reactions and animal populations grow or shrink exponentially.