Direct and inverse proportion
Twice as many apples cost twice as much. Picture, table and graph show together when a relation is proportional, with the rule of three.
MathematicsGrade 6–13 · University
Functions
From the concept of a function to linear, quadratic, rational, sine, exponential and polynomial functions – with sliders and draggable points.
Link to the German educational standards (KMK)
- L3Structures and functional relationships
- K3Modelling mathematically
- K4Using mathematical representations
- K5Working with mathematical objects
Functions and relationships
Grade 6–13 · University
Function or not? Relations and their graphs
Exactly one y for every x? The arrow diagram, table and graph tell you straight away whether a relation is a function.
Function plotter with custom terms
Up to three of your own functions at once, with a table of values, zeros and intersections. The tool for anything that can be a graph.
Linear functions
Grade 7–9
Linear function
Change the slope and the y-intercept of a line and read off the slope triangle and the zero.
Rational functions
Grade 7–13
Hyperbolas and their asymptotes
The hyperbola gets ever closer to its asymptotes and yet never touches them. Shifting and stretching show what the parameters do.
Poles and asymptotes of rational functions
A zero or a pole, a hole or a leap to infinity? Drag the terms of the fraction and the graph shows the difference.
Quadratic functions
Grade 9–11
Quadratic function
Stretch, reflect and shift parabolas and read off the vertex and the zeros.
Parabola through three points
Drag three points and the matching parabola always passes through all of them. The equation is worked out step by step.
Optimisation problems with parabolas
Which enclosure has the largest area? Picture and graph show the maximum, first by trying, then by exact calculation.
Power functions
Grade 9–11
Power functions y = a · xⁿ
Even or odd, positive or negative: y = a · xⁿ changes with every exponent, and only a few points always stay the same.
Trigonometric functions
Grade 9–11
Unit circle
Sine, cosine and tangent on the unit circle, and how the sine curve grows out of it.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Exponential functions and logarithms
Grade 9–13
Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
Exponential growth and decay
Water lilies on a pond, compound interest and decay: exponential growth compared with linear growth.
The natural exponential function
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
Polynomial functions
Grade 10–11
Polynomial functions: zeros, multiplicity and end behaviour
Does the graph cross the x-axis or only touch it? The multiplicity of the zero decides, the highest power decides what happens far out.
Properties of functions
Grade 10–12
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
Limits and asymptotes
What happens to a graph when x gets very large? A point runs outwards and a magnifier shows how close it gets to the asymptote.
Inverse functions: reflecting in y = x
Reflect the graph in y = x and the inverse function appears. It only works if no function value occurs twice.