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The problem asks you to determine a quadratic function \( f(x) = ax^2 + bx + c \) whose graph has the vertex (Scheitelpunkt) \( S(-6 \mid -1) \) and passes through the point \( P(-5 \mid 6) \).

### Steps to solve:

1. **Use the vertex form of a quadratic function:**
   \[
   f(x) = a(x - h)^2 + k
   \]
   where \( (h, k) \) is the vertex. Since the vertex is \( S(-6 \mid -1) \), we substitute \( h = -6 \) and \( k = -1 \):
   \[
   f(x) = a(x + 6)^2 - 1
   \]

2. **Substitute the point \( P(-5 \mid 6) \):**
   This point lies on the graph, so we substitute \( x = -5 \) and \( f(x) = 6 \) into the equation:
   \[
   6 = a(-5 + 6)^2 - 1
   \]
   Simplifying the equation:
   \[
   6 = a(1)^2 - 1
   \]
   \[
   6 = a - 1
   \]
   Solving for \( a \):
   \[
   a = 7
   \]

3. **Write the quadratic function:**
   Now that we have \( a = 7 \), the quadratic function is:
   \[
   f(x) = 7(x + 6)^2 - 1
   \]

4. **Expand the equation (optional):**
   If you want to express it in standard form \( f(x) = ax^2 + bx + c \), you can expand the equation:
   \[
   f(x) = 7(x^2 + 12x + 36) - 1
   \]
   \[
   f(x) = 7x^2 + 84x + 252 - 1
   \]
   \[
   f(x) = 7x^2 + 84x + 251
   \]

### Final Answer:
\[
f(x) = 7x^2 + 84x + 251
\]

Would you like more details on any step, or do you have further questions?

### Related Questions:
1. How do you determine the vertex of a quadratic function from its general form?
2. What are the key characteristics of a parabola's graph?
3. How does the value of \( a \) affect the direction and width of a parabola?
4. Can you solve for a quadratic function if given two points instead of a vertex?
5. What is the process of completing the square to convert a quadratic equation into vertex form?

### Tip:
Always remember that the vertex form of a quadratic equation is particularly useful when the vertex is known, as it simplifies the process of determining the equation!

Bestimme eine quadratische Funktion f(x) = ax^2 + bx + c, deren Graph den Scheitelpunkt S(-6 | -1) hat und durch den Punkt P(-5 | 6) verläuft.

Learn how to determine a quadratic function given its vertex at (-6, -1) and a point (-5, 6) on the graph. This problem includes key algebraic concepts, such as quadratic functions and the vertex form of equations. Suitable for students in Grades 9-11.

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