To design a roller coaster track using the equations given in the image, we need to:

1. Identify the existing segments provided in Part A.
2. Create additional equations that join these segments smoothly.
3. Ensure continuity at each joint to maintain smooth transitions.

Part A: Existing Track Segments

From the image, we observe three track segments:

1. Linear Segment $L_1$: $y = 0.7x$
2. Quadratic Segment $f(x)$: $f(x) = -0.078125x^2 + 0.7x$
3. Linear Segment $L_2$: $y = -1.8x + 20$

Each of these segments represents part of the roller coaster track.

Plan for Additional Segments

To complete the track, we need to add:

1. A section on the left of $L_1$ that smoothly connects with $L_1$ (entrance).
2. A section on the right of $L_2$ that extends to the exit.

1. Design Equations for the Entrance Section

To create a smooth connection before $L_1$, we can introduce a new linear or quadratic function that connects to $y = 0.7x$ with continuity in both value and slope at the transition point.

Assume the entrance is at $x = -5$. We’ll create two segments for the entrance:

Entrance Segment 1 (Parabolic)

Let's define this segment as: $y = a_1x^2 + b_1x + c_1$

For smoothness, we need it to satisfy:

• $y(-5) = y$ at the point where it meets $L_1$.
• ( y'(-5) = 0.