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The problem involves finding a parabolic function that models the cross-section of a bent strip and determining the slopes at the endpoints.

### Given:
- The plastic strip is placed on two wedges with a distance of 8 cm between them.
- The strip bends, forming a parabolic shape.
- Let \( h \) be the height of the parabola at its lowest point.

### a) **Finding the parabolic function:**

We can model the parabola using the standard form of a quadratic function:
\[
y(x) = ax^2 + bx + c
\]
Given that the parabola is symmetric and centered at the origin, we can simplify this to:
\[
y(x) = ax^2 + c
\]
The parabola passes through the points where \( x = \pm 4 \) cm (since the total width is 8 cm). At these points, \( y(x) = 0 \). Thus, we have:
\[
0 = a(4)^2 + c = 16a + c
\]
Also, at \( x = 0 \), the parabola reaches its maximum height \( y = h \):
\[
h = c
\]
So, substituting \( c = h \) in the previous equation:
\[
0 = 16a + h
\]
Solving for \( a \), we get:
\[
a = -\frac{h}{16}
\]
Thus, the equation of the parabola is:
\[
y(x) = -\frac{h}{16}x^2 + h
\]

### b) **Determining the slopes at the endpoints:**

The slope of the parabola at any point is given by the derivative of \( y(x) \):
\[
y'(x) = -\frac{h}{8}x
\]
At the endpoints \( x = \pm 4 \):
\[
\text{Slope at } x = 4: \quad y'(4) = -\frac{h}{8}(4) = -\frac{h}{2}
\]
\[
\text{Slope at } x = -4: \quad y'(-4) = -\frac{h}{8}(-4) = \frac{h}{2}
\]

### LaTeX solution:

```latex
\textbf{a) Parabelfunktion:} \\
y(x) = -\frac{h}{16}x^2 + h

\textbf{b) Steigungen an den Auflagepunkten:} \\
\text{Für } x = 4 \text{ ist die Steigung } y'(4) = -\frac{h}{2} \\
\text{Für } x = -4 \text{ ist die Steigung } y'(-4) = \frac{h}{2}
```

Would you like further details or have any questions?

Here are some related questions you might consider:
1. How would the parabola change if the distance between the wedges were increased?
2. What would happen to the parabola if it were symmetric but not centered at the origin?
3. How does the height \( h \) affect the curvature of the parabola?
4. Can you derive the focus and directrix of the parabola from the given equation?
5. How would the results change if the strip were placed on wedges that are not equidistant from the origin?

**Tip:** Remember that the vertex form of a parabola is often the most convenient when dealing with symmetry and specific points of the curve.

Explore how to model a parabolic function that represents the cross-section of a bent plastic strip supported by wedges. This problem involves finding the parabolic equation, determining slopes at endpoints, and understanding the impact of height on the curve's shape.

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