Detailed Solutions for Function Family Analysis

This page provides detailed solutions for the exercises introduced on the previous page. It focuses on the first exercise, which involves analyzing the function family f(x) = x² - (a + 1)x + a.

The solution process is broken down into several steps:

1. Calculating the first and second derivatives of the function
2. Finding the roots of the function using the quadratic formula
3. Determining the extrema of the function

Vocabulary: The quadratic formula is used to find the roots of a quadratic equation in the form ax² + bx + c = 0.

The solution demonstrates how to apply the quadratic formula to find the roots of the function:

x = (a + 1) ± √((a + 1)² - 4a) / 2

Highlight: Understanding how to find roots and extrema is crucial for analyzing the behavior of Funktionsscharen.

The page also includes a reminder about binomial formulas, which are useful in simplifying algebraic expressions encountered in these types of problems.

Example: The binomial formula (a + b)² = a² + 2ab + b² is used in simplifying expressions derived from the quadratic formula.

This detailed approach helps students understand the step-by-step process of analyzing function families and prepares them for more complex problems.

Graphing and Special Cases in Function Families

This page continues the analysis of the function family f(x) = x² - (a + 1)x + a, focusing on graphing and identifying special cases within the family.

Key points covered on this page include:

1. Graphing multiple functions from the family for different parameter values
2. Identifying the curve with a local extremum at x = 2
3. Finding the curve with exactly one root
4. Calculating an antiderivative of the function family

Example: The solution shows that the curve with a local extremum at x = 2 corresponds to a = 3.

The page demonstrates how to use the first and second derivative tests to determine the nature of extrema:

1. First derivative test: f'(x) = 0
2. Second derivative test: f''(x) > 0 for a minimum, f''(x) < 0 for a maximum

Highlight: Understanding how to identify special cases within a Funktionsschar is crucial for solving more complex problems involving function families.

The solution also covers finding an antiderivative (or indefinite integral) of the function family, which is an important skill for calculus applications.

Vocabulary: An antiderivative or Stammfunktion is a function F(x) whose derivative is the given function f(x).

This page provides a comprehensive look at how to analyze and interpret different aspects of a function family, building on the skills introduced in previous sections.

Analyzing Function Families: Quadratic Functions with Parameters

This page introduces two exercises focused on analyzing function families, specifically quadratic functions with parameters. These exercises are designed to help students understand the properties and behavior of Funktionsscharen (function families).

Definition: A function family, or Funktionsschar in German, is a set of functions that share a common form but differ by a parameter.

The exercises presented here cover various aspects of function analysis, including:

1. Finding roots and extrema
2. Sketching graphs for different parameter values
3. Identifying specific curves with certain properties
4. Calculating areas under curves

Highlight: These exercises provide a comprehensive approach to understanding how parameters affect the behavior of quadratic functions.

The problems are structured to gradually increase in complexity, allowing students to build their skills and understanding of Funktionsscharen step by step.

Example: The first exercise introduces the function family f(x) = x² - (a + 1)x + a, where a ≥ 1 and a is a real number.

This type of problem is excellent for developing analytical skills and gaining a deeper understanding of quadratic functions and their properties.