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.
Mathematics – grade 11
Introductory phase of upper secondary (G9: grade 11). According to the core curriculum, the order and scope of the three learning areas are a suggestion; the sub-areas are the items of the “core”. The codes are our own.
Unit EF-1Descriptive statistics
Collecting and grouping data, bar charts and measures such as mean, median, mode, range, variance and standard deviation.
EF-1.1Collecting data
Characteristics, grouping, representativeness and bar charts.
No separate simulation is planned for this.
EF-1.2Measures
Interpreting measures of centre and spread and comparing data sets; optionally box plots.
No separate simulation is planned for this.
Unit EF-2Elementary functions
Power and root functions, comparing parameter variation for power, exponential and sine functions, and polynomial functions with zeros, symmetry and end behaviour.
EF-2.1Power functions
Sketching graphs, end behaviour and symmetry, root functions.
EF-2.2Comparing power, exponential and sine functions
Parameter variation a · f(b · (x − c)) + d for different classes of functions.
Shifting, stretching and reflecting graphs
Shift, stretch, reflect: every move of the graph leaves its mark in the function's equation.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
EF-2.3Polynomial functions
Seen as sums of power functions, modelling, zeros and factorised form, end behaviour and symmetry.
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.
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.
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.
EF-2.4Optional extensions
The root function and the reciprocal function as inverse functions.
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.
Unit EF-3Derivatives
From average to instantaneous rate of change, the derivative function graphically and with rules, tangents, monotonicity, extrema and points of inflection, and optimisation problems.
EF-3.1Derivative at a point
Difference quotient, slope of secant and tangent, limit notation.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
EF-3.2Derivative function
Developing graph and derivative graph from each other, sum and constant-multiple rules, derivatives of power, sine and cosine functions.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
EF-3.3Using derivatives
Tangents and normals, monotonicity, extrema and points of inflection, optimisation problems.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 11