An amusement park is building a roller coaster with a drop section modeled by a quadratic. The roller coaster will dip 10 feet below ground level. The roller coaster will dip below ground level at a horizontal distance of 32 feet from the peak and re-emerge to ground level at a horizontal distance of 72 feet from the peak.

What is the expression that models this scenario?

To model the drop section of the roller coaster using a quadratic function, we can define a quadratic equation of the form:

\[ y = ax^2 + bx + c \]

where \( y \) represents the height of the roller coaster relative to ground level, and \( x \) represents the horizontal distance from the peak.

Given the conditions:
1. The roller coaster dips 10 feet below ground level. Thus, the minimum height is \( y = -10 \).
2. The roller coaster dips below ground level at a horizontal distance of 32 feet from the peak. This means when \( x = 32 \), \( y = 0 \).
3. The roller coaster re-emerges to ground level at a horizontal distance of 72 feet from the peak. This means when \( x = 72 \), \( y = 0 \).

### Step 1: Set up the equation
Let's write the quadratic equation in vertex form, which is:

\[
y = a(x - h)^2 + k
\]

where \( (h, k) \) is the vertex of the parabola.

### Step 2: Identify the vertex
The vertex \( (h, k) \) represents the lowest point of the parabola. Here, the roller coaster dips 10 feet below ground level, so \( k = -10 \), and this occurs at the midpoint between the two points where the roller coaster crosses ground level.

The midpoint \( h \) is:

\[
h = \frac{32 + 72}{2} = \frac{104}{2} = 52 \text{ feet}
\]

Thus, the vertex is \( (52, -10) \).

### Step 3: Write the equation in vertex form
The quadratic function can now be written as:

\[
y = a(x - 52)^2 - 10
\]

### Step 4: Determine the value of \( a \)
We know that when \( x = 32 \) and \( x = 72 \), \( y = 0 \). Using \( x = 32 \) in the equation:

\[
0 = a(32 - 52)^2 - 10
\]

\[
0 = a(-20)^2 - 10
\]

\[
0 = 400a - 10
\]

\[
400a = 10
\]

\[
a = \frac{10}{400} = \frac{1}{40}
\]

### Final Expression
Substitute \( a = \frac{1}{40} \) back into the vertex form equation:

\[
y = \frac{1}{40}(x - 52)^2 - 10
\]

This is the expression that models the roller coaster's drop section.

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Would you like further details or have any questions?

Here are some related questions:
1. How would the equation change if the roller coaster dipped 20 feet below ground instead?
2. What would be the impact on the equation if the roller coaster re-emerged at 82 feet instead of 72 feet?
3. How can you determine the horizontal distance from the peak to any given height?
4. What is the general method for converting from vertex form to standard form for a quadratic equation?
5. How would the graph of this equation look like if the roller coaster's peak height was changed?

**Tip:** When dealing with quadratic functions, the vertex form is especially useful when you know the vertex (highest or lowest point) of the parabola.

Learn how to model a roller coaster's drop section using a quadratic equation. This problem involves determining the quadratic equation for a roller coaster that dips 10 feet below ground at 32 feet from the peak and re-emerges at 72 feet from the peak. Explore the vertex form and key calculations for this scenario.

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