To analyze the transformation of the polynomial function representing the roller coaster, let's break down the expression $2 f(0.3x - 1) + 10$.

1. Horizontal Compression: The term $0.3x$ compresses the graph horizontally by a factor of $\frac{1}{0.3} \approx 3.33$. This means that for every $x$ value, the corresponding $y$ value will occur sooner on the x-axis.

2. Horizontal Shift: The term $-1$ shifts the graph to the right by 1 unit. This means that the entire graph is moved one unit to the right.

3. Vertical Stretch and Shift: The factor of $2$ means the $y$ values are multiplied by 2, which stretches the graph vertically. The $+10$ shifts the entire graph up by 10 units.

Now let's analyze how these transformations affect the specific features of the function:

• Original y-intercept: The original function crosses the y-axis at $(0, 4)$. After transformation: $y = 2 \cdot 4 + 10 = 8 + 10 = 18.$
• x-intercepts: Originally, the function crosses the x-axis at $(5, 0)$ and $(8, 0)$. After the transformations, the new x-intercepts will be calculated based on the horizontal compression and shift.

For a more detailed exploration of the transformed function's behavior:

1. Y-Intercept: $(0, 18)$
2. New behavior near the x-intercepts: The new x-intercepts will not be exactly $5$ and $8$ but will shift based on the compression and right shift.

Comparing with the given options, the transformation primarily results in a function that crosses the y-axis higher and shifts up. Therefore, let's examine the given curves:

1. The first curve goes from (0, 10) and behaves as described above.
2. The fourth curve matches the new shape, with significant shifts and crossings.

Based on the transformations, the best representation of the modified roller coaster design is the fourth option, where it starts from (0, 10), decreases, and then increases through (20, -10) and (30, -10).

Summary

The correct answer is the fourth option.

Would you like more details on the transformations or any specific aspects? Here are five related questions to consider:

1. What specific characteristics of the original polynomial function can we identify?
2. How do different transformations affect the shape of polynomial graphs?
3. What would happen if we applied a negative coefficient instead of a positive one?
4. How can we determine the new x-intercepts after the transformations?
5. What role does the degree of the polynomial play in these transformations?

Tip: When analyzing transformations, sketch the original and transformed functions to visualize the changes clearly.