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This problem involves a parabolic curve connecting two sections of a railway, with specific slopes at each end. Here's the analysis of each part of the question:

### Problem Summary
- **Task**: Find the parameters of the parabolic function \( f(x) = ax^2 + bx + c \).
- The railway has a slope of 50% at the beginning and -100% at the end.
- The horizontal distance between points A and B is 100 m.

### Problem Breakdown

1. **(a) Determine \( a \), \( b \), and \( c \)**:
   - The curve needs smooth transitions at points A and B.
   - The slope at A is 50% (0.5 in gradient form), and the slope at B is -100% (-1.0 in gradient form).
   - Use the coordinates of A and B, and their respective slopes, to find the coefficients.

2. **(b) Calculate the height difference between A and B**:
   - Use the parabolic function found in part (a) to determine the vertical positions of A and B.

3. **(c) Determine the highest point of the parabola and its height above point A**:
   - The highest point corresponds to the vertex of the parabola.

Let's start solving the problem by calculating the coefficients of the parabola using the given conditions. 

### Step-by-Step Solution

#### 1. Set Up Equations for \( a \), \( b \), and \( c \)

Given that:
- At \( x = 0 \) (point A), the slope is 0.5.
- At \( x = 100 \) m (point B), the slope is -1.0.
- Assume point A is at the origin \((0, 0)\).

The conditions translate to:
- \( f'(0) = 0.5 \).
- \( f'(100) = -1.0 \).

Let’s solve these equations mathematically.The coefficients of the parabolic function \( f(x) = ax^2 + bx + c \) are:

- \( a = -0.0075 \)
- \( b = 0.5 \)
- \( c = 0 \)

Thus, the parabolic equation connecting points A and B is:

\[
f(x) = -0.0075x^2 + 0.5x
\]

### Next Steps:
Now let's solve parts (b) and (c):

1. **(b) Height difference between A and B**:
   - Find the values of \( f(100) \) and \( f(0) \).

2. **(c) Highest point of the parabola and its height above point A**:
   - Determine the vertex of the parabola, which is the highest point, and find its y-coordinate.

Let's calculate these values.### Results:

1. **Height difference between A and B**:
   - Height at A: \( 0 \) m
   - Height at B: \(-25\) m
   - Height difference: \( -25 \) m

2. **Highest point of the parabola (vertex)**:
   - The highest point occurs at \( x = 33.33 \) m.
   - Height of the highest point: \( 8.33 \) m
   - Height above point A: \( 8.33 \) m

### Additional Questions:
1. How does changing the slope at points A and B affect the shape of the parabola?
2. What would happen to the height difference if the slopes were reversed?
3. How would the highest point shift if the distance between A and B were increased?
4. How can you find the x-coordinate of the highest point for any general quadratic function?
5. What other shapes (besides parabolas) can be used for smooth transitions between two slopes?

**Tip**: The vertex formula \(-\frac{b}{2a}\) is a quick way to find the highest or lowest point of a quadratic function, depending on whether the parabola opens upwards or downwards.

A mountain railway has a straight incline of 50% and a straight descent of -100%, with a parabolic connection between the two sections. The parabolic equation is f(x) = ax^2 + bx + c. Determine a, b, and c to ensure smooth transitions at points A and B, calculate the height difference, and find the highest point of the parabola.

Explore a mountain railway problem involving a parabolic curve with given slopes of 50% and -100%. Learn how to calculate the quadratic equation's coefficients, determine the height difference between points A and B, and find the highest point of the parabola. Ideal for Grades 10-12 students.

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