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

### Step 1: Use the Vertex Form
The vertex form of a quadratic function is:

\[
f(x) = a(x - h)^2 + k
\]

where \( (h, k) \) is the vertex. Given that the vertex is \( S(-6, -1) \), we substitute these values into the vertex form:

\[
f(x) = a(x + 6)^2 - 1
\]

### Step 2: Use the Point \( P(-5, 6) \)
We know that the function passes through the point \( P(-5, 6) \), meaning \( f(-5) = 6 \). We substitute \( x = -5 \) and \( f(x) = 6 \) into the equation:

\[
6 = a(-5 + 6)^2 - 1
\]

Simplify the equation:

\[
6 = a(1)^2 - 1
\]
\[
6 = a - 1
\]

Solve for \( a \):

\[
a = 7
\]

### Step 3: Final Equation
Now that we know \( a = 7 \), we substitute it back into the equation:

\[
f(x) = 7(x + 6)^2 - 1
\]

### Step 4: Convert to Standard Form (Optional)
To express the function in the form \( f(x) = ax^2 + bx + c \), expand the equation:

\[
f(x) = 7(x^2 + 12x + 36) - 1
\]
\[
f(x) = 7x^2 + 84x + 252 - 1
\]
\[
f(x) = 7x^2 + 84x + 251
\]

Thus, the quadratic function is:

\[
f(x) = 7x^2 + 84x + 251
\]

Let me know if you would like more details or explanations.

### Here are 5 related questions to explore:
1. How do you derive the vertex form of a quadratic equation?
2. Can you explain how the vertex form relates to transformations of the basic parabola?
3. What role does the value of \( a \) play in the shape of the parabola?
4. How would the function change if the vertex was at a different point?
5. How can we find the x-intercepts of this quadratic function?

### Tip:
When solving quadratic problems with vertex and another point, always use the vertex form for simpler substitutions.

Determine a quadratic function f(x) = ax^2 + bx + c whose graph has the vertex S(-6, -1) and passes through the point P(-5, 6).

Find the quadratic function with the vertex S(-6, -1) that passes through the point P(-5, 6). This solution involves applying the vertex form of a quadratic equation and converting to standard form. Suitable for Grades 9-11.

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