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The task is to write the equation of the parabola in vertex form based on the provided graph from the amusement park ride, where the vertex appears to be at the point \( (0, 180) \) and another point on the parabola is \( (1, 164) \).

### Vertex Form of a Parabola:
The vertex form of a quadratic equation is:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex.

From the graph:
- The vertex is \( (0, 180) \), so \( h = 0 \) and \( k = 180 \).
- We now have the equation:
\[
y = a(x - 0)^2 + 180
\]
which simplifies to:
\[
y = a x^2 + 180
\]

### Finding the value of \( a \):
To find \( a \), we can use the second point given on the graph: \( (1, 164) \).

Substitute \( x = 1 \) and \( y = 164 \) into the equation:
\[
164 = a(1)^2 + 180
\]
Simplifying:
\[
164 = a + 180
\]
\[
a = 164 - 180
\]
\[
a = -16
\]

### Final Equation:
Substituting \( a = -16 \) into the vertex form equation:
\[
y = -16x^2 + 180
\]

This is the equation of the parabola in vertex form.

Would you like more details or have any questions?

Here are five related questions you might want to explore:
1. How do you convert a parabola from standard form to vertex form?
2. What is the significance of the \( a \)-value in the vertex form of a parabola?
3. How can you find the axis of symmetry from the vertex form of a parabola?
4. Can you calculate the focus and directrix of this parabola?
5. How would you graph a parabola given only its equation in vertex form?

**Tip**: The \( a \)-value in the vertex form determines how "narrow" or "wide" the parabola is, and its sign tells you whether the parabola opens upwards or downwards.

Write an equation of the parabola in vertex form based on the amusement park ride graph.

Learn how to write the equation of a parabola in vertex form using a graph from an amusement park ride. The problem uses points (0, 180) and (1, 164) to determine the equation in the form y = a(x - h)^2 + k. Suitable for students in Grades 9-10.

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