Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Mathematics – grade 12/13
Qualification phase (G9: grades 12 and 13). The core curriculum has separate learning areas for the basic (gA) and the higher requirement level (eA) and does not distribute them over the years, so they are the same for the double grade 12 and 13. The codes are our own.
Unit gA-1Curve fitting with polynomial functions (basic level)
Turning properties into conditions on the function term, solving linear systems algorithmically and adjusting a parameter to a given property.
Unit gA-2From change to total – integral calculus (basic level)
Reconstructing totals from rates of change, the integral as a limit of Riemann sums, the fundamental theorem, areas bounded by graphs and antiderivatives.
gA-2.1Definite integral
Reconstructed totals, Riemann sums, the fundamental theorem and areas.
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.
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.
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.
gA-2.2Antiderivative
Stating, developing and checking antiderivatives with the rules of differentiation.
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.
Unit gA-3The exponential function e^x (basic level)
The base e via (eˣ)′ = eˣ, combining and composing with other functions, product and chain rules, exponential equations and limited 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.
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.
Unit gA-4Spatial visualisation and coordinates (basic level)
Points and vectors in space, vector operations, lines and planes in parametric form, distances between points, the dot product, angles and relative positions of lines.
gA-4.1Spatial visualisation and coordinates
Points and vectors as tuples, addition and scalar multiplication, collinearity, lines and planes in parametric form.
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.
Lines and planes in space
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
gA-4.2Measures and positions
Distances between points, dot product and orthogonality, angles, relative positions and intersections of lines.
Relative positions and distances in space
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
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.
gA-4.3Optional extensions
Lines and planes, normal form, equations of circles and spheres.
Spheres, lines and planes
A plane cuts the sphere in a circle, touches it at a point or misses it entirely. Including the equation of the sphere and the tangent plane.
Unit gA-5Data and chance (basic level)
Conditional probability and independence, expected value and standard deviation, the binomial distribution with prediction intervals and simulations.
gA-5.1Conditional probability and stochastic independence
Developing it with tree diagrams and two-way tables, checking stages for independence.
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.
Monty Hall problem
Play yourself or simulate thousands of games: is it worth switching doors?
gA-5.2Expected value and standard deviation of discrete random variables
Calculating and interpreting measures, fair games.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
gA-5.3Binomial distribution
Judging the model, calculating probabilities and measures, interpreting histograms, prediction intervals and whether a sample is compatible.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Unit eA-1From change to total – integral calculus (higher level)
As at basic level, plus the integral function, ln as an antiderivative of 1/x, solids of revolution and improper integrals.
eA-1.1Definite integral
Reconstructed totals, Riemann sums, the fundamental theorem and areas.
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.
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.
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.
eA-1.2Integral function and antiderivative
The integral function as accumulation or area function, antiderivatives including ln.
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.
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.
eA-1.3Further topics
Volumes of solids of revolution about the x-axis, improper integrals.
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 eA-2Growth models – exponential function (higher level)
Comparing limited and logistic growth, differential equations as models, the exponential function with its derivative, combinations and compositions, and exponential equations.
eA-2.1Investigating growth processes
Limited and logistic growth, asymptotic behaviour, differential equations.
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.
eA-2.2Exponential function e^x
The base e, derivatives, product and chain rules, exponential equations and family parameters.
The natural exponential function
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
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.
Unit eA-3Spatial visualisation and coordinates (higher level)
As at basic level, plus projection onto the plane with matrices, planes in normal and coordinate form, distances between points, lines and planes, and Gaussian elimination.
eA-3.1Spatial visualisation and coordinates
Points and vectors, vector operations, collinearity and oblique drawings with projection matrices.
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.
eA-3.2Forms of representation
Lines and planes in parametric, normal and coordinate form.
Lines and planes in space
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
eA-3.3Measures and positions
Distances, dot product, angles, relative positions of lines and planes, and Gaussian elimination.
Relative positions and distances in space
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
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.
eA-3.4Optional extensions
Vectors in other contexts, further transformation matrices, equations of circles and spheres.
Matrices as linear maps
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
Spheres, lines and planes
A plane cuts the sphere in a circle, touches it at a point or misses it entirely. Including the equation of the sphere and the tangent plane.
Unit eA-4Curve fitting and families of functions (higher level)
Classifying functions by global properties and fitting them to data, piecewise functions, and families of polynomial and exponential functions.
eA-4.1Curve fitting
Using symmetry, behaviour for large |x|, asymptotes and periodicity, setting up conditions, continuity and differentiability.
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.
eA-4.2Families of functions
The effect of the family parameter and fitting to properties.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Unit eA-5Data and chance (higher level)
As at basic level, plus causal and stochastic independence, the normal distribution, approximating the binomial distribution and confidence intervals.
eA-5.1Conditional probability and stochastic independence
Tree diagrams and two-way tables, causal and stochastic independence.
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.
Monty Hall problem
Play yourself or simulate thousands of games: is it worth switching doors?
eA-5.2Expected value and standard deviation of discrete random variables
Calculating and interpreting measures, fair games.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
eA-5.3Binomial distribution
The model, probabilities, measures, histograms, prediction intervals and simulations.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
eA-5.4Normal distribution
Discrete and continuous random variables, parameters of the normal distribution.
Normal distribution
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
eA-5.5Binomial and normal distribution
Approximating the binomial distribution, sigma intervals and confidence intervals.
Normal distribution
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 12/13