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To write the equation of the parabola in vertex form, we can use the general vertex form of a quadratic equation:

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

Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) is a coefficient that determines the direction and width of the parabola.

From the graph:
- The vertex is \( (0, 180) \), so \( h = 0 \) and \( k = 180 \).
- Another point on the graph is \( (1, 164) \), which we can substitute into the equation to solve for \( a \).

Starting with the equation:
\[
y = a(x - 0)^2 + 180 \quad \text{or} \quad y = ax^2 + 180
\]

Now, substitute \( (x, y) = (1, 164) \):
\[
164 = a(1)^2 + 180
\]
\[
164 = a + 180
\]
\[
a = 164 - 180 = -16
\]

Thus, the equation of the parabola in vertex form is:
\[
y = -16x^2 + 180
\]

Would you like more details on this process or have any questions?

Here are five related questions:
1. How would the equation change if the vertex was different?
2. Can you convert this vertex form into standard form?
3. How do you determine if a parabola opens upward or downward?
4. What is the role of the coefficient \(a\) in transforming the shape of the parabola?
5. How can you find the x-intercepts from this vertex form?

**Tip:** When finding the equation of a parabola, always make sure you correctly identify the vertex and use another point to solve for the constant \(a\).

Write an equation of the parabola in vertex form based on the graph showing a ride's height over time.

This problem involves finding the equation of a parabola given a graph of a ride's height over time. It demonstrates how to use the vertex form of a quadratic equation y = a(x - h)^2 + k, and how to substitute another point to determine the constant 'a'. The problem is suitable for students in Grades 9-11.

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