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Let's focus on problem **#10 (grün)** from the image you provided, which asks to solve the following:

**Task:**
- Two upward-opening parabolas, \( p_1 \) and \( p_2 \), have the following vertices:
  - \( S_1(-1, 3) \) for \( p_1 \)
  - \( S_2(4, 2) \) for \( p_2 \)

**Subtasks:**
a) Calculate the equations of both parabolas.
b) Calculate the coordinates of their intersection points.
c) Show that the points \( S_1 \), \( S_2 \), and \( Q \) form a triangle and determine its perimeter.

### Step 1: Find the equations of the parabolas
Both parabolas are vertically shifted, so their general form is:

\[
p(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex.

- **For \( p_1 \)**:
  - Vertex \( S_1(-1, 3) \)
  - Equation of \( p_1 \): \( p_1(x) = a_1(x + 1)^2 + 3 \)
  
- **For \( p_2 \)**:
  - Vertex \( S_2(4, 2) \)
  - Equation of \( p_2 \): \( p_2(x) = a_2(x - 4)^2 + 2 \)

To find the values of \( a_1 \) and \( a_2 \), we need more information or to proceed by assuming standard forms (e.g., \( a_1 = a_2 = 1 \)) unless otherwise specified.

Let me calculate these in detail next.The intersection point between the parabolas \( p_1(x) = (x + 1)^2 + 3 \) and \( p_2(x) = (x - 4)^2 + 2 \) occurs at \( x = \frac{7}{5} \).

### Step 2: Calculate the y-coordinate of the intersection point
We can now substitute \( x = \frac{7}{5} \) into either equation (both should yield the same \( y \)-value) to find the corresponding \( y \)-coordinate. Let me calculate that next.The \( y \)-coordinate of the intersection point is approximately \( 8.76 \). Thus, the coordinates of the intersection point \( Q \) are:

\[
Q\left( \frac{7}{5}, 8.76 \right)
\]

### Step 3: Determine if \( S_1(-1, 3) \), \( S_2(4, 2) \), and \( Q\left( \frac{7}{5}, 8.76 \right) \) form a triangle and calculate its perimeter.

To form a triangle, the points must be non-collinear. We will:
1. Check if they are collinear by calculating the slopes between the points.
2. If they are not collinear, calculate the perimeter using the distances between the points.

I'll proceed with these calculations now.The points \( S_1(-1, 3) \), \( S_2(4, 2) \), and \( Q\left( \frac{7}{5}, 8.76 \right) \) are not collinear and form a triangle. The perimeter of this triangle is approximately **18.58 units**.

Would you like further details on any of these steps?

### 5 Related Questions:
1. How do you find the equation of a parabola given its vertex and a point?
2. What is the geometric significance of the intersection of two parabolas?
3. How can you verify if three points are collinear using slopes?
4. How do you calculate the area of a triangle given three vertices?
5. Can parabolas with vertical shifts still intersect at more than one point?

### Tip:
When working with parabolas, always remember that translating the vertex horizontally or vertically doesn’t change the overall shape, but can affect how and where they intersect.

Solve Aufgabe 10 Grün: Find the equations of two parabolas, their intersection points, and verify if three specific points form a triangle, calculating its perimeter.

Explore the solution to finding the equations of two parabolas from their vertices, calculating their intersection points, and determining if three points form a triangle with a calculated perimeter. This task utilizes algebra and geometry principles suitable for high school students.

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