An engineer is using a polynomial function to model the height of a roller coaster over time x, as shown.
On a coordinate plane, a curve increases from quadrant 3 and crosses the y-axis at (0, 4). It then decreases and crosses the x-axis at (5, 0). It continues to decrease and then starts to increase and crosses the x-axis at (8, 0).
The engineer wants to modify the roller coaster design by transforming the function. Which represents 2 f (0.3 x minus 1) + 10, the modified design of the roller coaster?
On a coordinate plane, a curve increases from quadrant 3 and crosses the y-axis at (0, 10). It then decreases and goes through (20, 10). It continues to decrease and then starts to increase and goes through (30, 10).
On a coordinate plane, a curve increases from quadrant 3 and crosses the y-axis at (0, negative 10). It then decreases and goes through (20, negative 10). It continues to decrease and then starts to increase and goes through (30, negative 10).
On a coordinate plane, a curve increases from quadrant 3 and crosses the y-axis at (0, 20). It then decreases and then increases again.
On a coordinate plane, a curve increases from quadrant 3 and crosses the y-axis at (0, 10). It then decreases and goes through (20, negative 10). It continues to decrease and then starts to increase and goes through (30, negative 10).

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.

An engineer is using a polynomial function to model the height of a roller coaster over time x, as shown. On a coordinate plane, a curve increases from quadrant 3 and crosses the y-axis at (0, 4). It then decreases and crosses the x-axis at (5, 0). It continues to decrease and then starts to increase and crosses the x-axis at (8, 0). The engineer wants to modify the roller coaster design by transforming the function. Which represents 2 f (0.3 x - 1) + 10, the modified design of the roller coaster?

Explore how polynomial function transformations can model a roller coaster's height over time. This problem covers key concepts like horizontal compression, vertical stretching, and shifts, examining the transformation 2 f(0.3x - 1) + 10 and its impact on the function's graph. Ideal for students in Grades 10-12.

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