The worksheet appears to focus on quadratic functions and parabolas, specifically transformations, finding vertex points, and rewriting quadratic equations in different forms. Let me break down the key questions and how they should be approached:

1. Vertex and Zeros (Nullstellen)

• Task: Identify the vertex (Scheitelpunkt) and the number of zeros (Nullstellen) for each given function in the table. Then sketch the graphs.

Approach:

• For functions like $f(x) = (x + 2.5)^2$, the vertex form $f(x) = (x - h)^2 + k$ gives the vertex $(h, k)$. Here:
  • $h = -2.5$ and $k = 0$, so the vertex is $(-2.5, 0)$.
  • The number of zeros depends on whether $k > 0$, $k < 0$, or $k = 0$. For $k = 0$, there is 1 zero.

2. Rewriting Vertex Form to Standard Form

• Task: For each parabolic equation in vertex form, expand and rewrite it in the general quadratic form $f(x) = ax^2 + bx + c$.

Approach for Example (a):

• Start with $f(x) = (x + 3)^2 + 1.5$.
• Expand $(x + 3)^2 = x^2 + 6x + 9$, so: $f(x) = x^2 + 6x + 9 + 1.5 = x^2 + 6x + 10.5$

Repeat this for all other equations.

3. Points on the Graphs

• Task: Note specific points from the graphs of given functions.

Approach:

• Use the function definitions to calculate $y$ for given $x$-values.
• For $f(x) = (x + 1)^2 - 6$, substituting $x = -1$ gives $y = -6$, so $(-1, -6)$ is a point.

4. Finding Vertex from General Form

• Task: Convert equations from general form to vertex form to find the vertex.

Approach:

• Use completing the square. For $f(x) = x^2 + 4x + 1$:
  • Factor out 1 from the $x^2$ and $x$ terms: $f(x) = (x^2 + 4x) + 1$.
  • Add and subtract $(4/2)^2 = 4$: $f(x) = (x^2 + 4x + 4) - 4 + 1$.
  • Simplify to $f(x) = (x + 2)^2 - 3$.
  • The vertex is $(-2, -3)$.

Would you like me to go through each specific problem in detail or provide worked-out solutions for particular sections? Here are some related questions:

Relative Questions:

1. How does the vertex form relate to transformations of the graph?
2. How can you quickly determine the number of zeros from the vertex form?
3. What is the geometric meaning of completing the square?
4. How do changes in $k$ in $f(x) = (x - h)^2 + k$ affect the graph's position?
5. What are efficient methods to sketch quadratic functions without plotting many points?

Tip: Always check the symmetry of a parabola around its vertex to verify your graph.