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.
Mathematics – grade 11
Studienstufe, first year (S1/S2; at the Gymnasium with Abitur after grade 12: grade 11): modules 1.1, 1.2, 2 and 3. Module 1.2 may also be taught before module 4, and the binomial distribution from module 5 right after module 2. eA = higher level only. The numbers are the module numbers of the curriculum.
Unit 1.1From rate of change to total (analysis)
Polynomial, simple rational and root functions with parameters, symmetry, behaviour at infinity and asymptotes, rules of differentiation, curve sketching and optimisation, the integral, antiderivatives and the fundamental theorem; eA: families of functions, solids of revolution.
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.
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.
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.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Optimisation problems
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
From a graph to its antiderivative
The height of one graph becomes the slope of another. If you understand bank balances, the antiderivative comes quicker than you think.
The integral as signed area (rectangle sums)
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Integral function and fundamental theorem
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
The integral as total change: from velocity to distance
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Solids of revolution
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Unit 1.2From rate of change to total (trigonometric functions)
Sine and cosine functions with parameters in context, their derivatives and antiderivatives, the product and chain rules, areas.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
Unit 2Modelling chance (stochastics)
Relative frequency and probability, measures of centre and spread with variance and standard deviation, tree diagrams and two-way tables, stochastic independence, simulations.
Two-way tables and Venn diagrams
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Conditional probability with medical tests
A medical test is positive, but how likely is it that you are really ill? 1000 people show why the answer is often surprising.
Simulating the birthday problem
How likely is it that two people in a class share a birthday? From just 23 people on, it is more likely than not.
Monty Hall problem
Play yourself or simulate thousands of games: is it worth switching doors?
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Unit 3Coordinate geometry
Vectors in the plane and in space, position vectors, points and objects in the coordinate system, calculating with vectors; eA: magnitude and dot product.
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Adding and scaling vectors (3D)
Drag vector arrows in space, add them and multiply them by numbers. The calculation is shown alongside in column form.
Dot product and cross product
The dot product tells you the angle between two vectors; the cross product gives a perpendicular vector and an area.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Bildungsplan, Gymnasium, Mathematics, grade 11