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.
Mathematics – grade 11
Introductory phase E1/E2 “Foundations of calculus”. Topic areas E.1–E.5 are mandatory. The core curriculum only names course half-years; assigning them to grade 11 is an assumption (G9).
Unit E.1Functions and their representation
Mandatory topic area. Revisiting and deepening the concept of function using polynomial functions: properties, special points and modelling with linear and quadratic functions.
E.1.1Developing basic concepts and properties using polynomial functions
Domain and range, symmetry, behaviour for large |x|, and shifting and stretching graphs.
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.
E.1.2Determining special points
Finding axis intercepts and intersections of two graphs graphically and algebraically.
No separate simulation is planned for this.
E.1.3Modelling real situations with suitable classes of functions
Describing real situations mainly with linear and quadratic functions.
No separate simulation is planned for this.
Unit E.2Introduction of the derivative
Mandatory topic area. From average to instantaneous rate of change: secant becomes tangent, derivative at a point and derivative function, first differentiation rules.
E.2.1Meaning of the derivative as instantaneous rate of change and as slope of the tangent
Interpreting rates of change in context, especially average and instantaneous velocity.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
E.2.2Derivative of a function at a point
Limit from difference quotient to differential quotient using an intuitive notion of limit; tangent and normal.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
E.2.3Relationship between graphs of a function and its derivative
Differentiating graphically, matching graphs of f and f′ with reasons; concept of an antiderivative.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
E.2.4Differentiation rules
Justifying the power, constant-factor and sum rules and differentiating polynomials.
No separate simulation is planned for this.
Unit E.3Applications of the derivative
Mandatory topic area. Curve sketching of polynomial functions, reconstructing functions from properties and solving optimisation problems.
E.3.1Analysis of polynomial functions
Determining monotonicity, concavity, extrema and inflection points with necessary and sufficient conditions.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
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.
E.3.2Reconstructing polynomial functions from their properties
Finding a suitable function equation from conditions (e.g. extreme point, smooth join).
No separate simulation is planned for this.
E.3.3Optimisation problems
Solving optimisation problems inside and outside mathematics with the derivative.
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.
Unit E.4Exponential functions
Mandatory topic area. Exponential growth and decay, the natural exponential function and its derivative, product and chain rule, and solving exponential equations.
E.4.1Growth and decay processes
Interpreting parameters in f(x) = a·bˣ + c, half-life and doubling time, modelling with data.
No separate simulation is planned for this.
E.4.2The (natural) exponential function f(x) = eˣ
Deriving the number e and the derivative of eˣ intuitively; changing to base e.
No separate simulation is planned for this.
E.4.3Product and chain rule
Differentiating products and compositions of polynomial and exponential functions.
No separate simulation is planned for this.
E.4.4Exponential equations
Solving exponential equations algebraically using logarithms.
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 E.5Trigonometric functions
Mandatory topic area. Describing periodic processes with sine and cosine functions, interpreting parameters and finding the derivatives of these functions.
E.5.1Periodic processes
Radians, meaning of the parameters in a·sin[b·(x − c)] + d, period, amplitude and modelling with data.
General sine function
Change the amplitude, period and shift of a sine curve with sliders or by dragging.
E.5.2Derivative of the sine and cosine functions
Justifying the derivatives by graphical differentiation.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
E.5.3Derivative of f(x) = a·sin[b·(x − c)] + d or f(x) = a·cos[b·(x − c)] + d
Differentiating with the chain rule and interpreting derivative values in context; tangents and normals.
No separate simulation is planned for this.
Unit E.6Further methods for solving equations
Optional topic area. Polynomial division to find zeros and numerical approximation methods for equations.
E.6.1Polynomial division
Factorising polynomials to determine zeros.
No separate simulation is planned for this.
E.6.2Numerical solution methods
Comparing bisection, Newton's method and regula falsi, including convergence and limitations.
Newton’s method
Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.
Unit E.7Sequences and series
Optional topic area. Arithmetic and geometric sequences and series and their limits.
E.7.1Sequences
Justifying convergence and divergence and finding limits with limit laws.
No separate simulation is planned for this.
E.7.2Series
Investigating arithmetic and geometric series and their limits.
No separate simulation is planned for this.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 11