https://file.aiquickdraw.com/mathbot/file/1728832748260q7e8wsaf.jpg

Let's address each part of the problem step by step.

### Given Information:
The equation of the curve is:
\[
H = a(x - h)^2 + k
\]
where:
- \( H \) is the height in meters.
- \( x \) is the distance from the starting point.
- The \( y \)-intercept is \((0, 19)\).
- The vertex is at \((3, 1)\).
- Point \( P(5, 9) \) is on the graph.

### Solution:

#### a. Starting Height
From the graph, the roller coaster journey begins at the \( y \)-intercept \((0, 19)\). Therefore, the starting height is:
\[
H = 19 \text{ meters}
\]

#### b. Values of \( h \) and \( k \)
Since the vertex of the curve is at \((3, 1)\), this gives us:
\[
h = 3 \quad \text{and} \quad k = 1
\]

#### c. Equation of the Curve
Now that we have \( h = 3 \) and \( k = 1 \), the equation becomes:
\[
H = a(x - 3)^2 + 1
\]
We need to determine the value of \( a \). We can use the point \( (0, 19) \), which is on the graph:
\[
19 = a(0 - 3)^2 + 1
\]
\[
19 = 9a + 1
\]
\[
18 = 9a
\]
\[
a = 2
\]
Thus, the equation of the curve is:
\[
H = 2(x - 3)^2 + 1
\]

#### d. Verification that \( a = 2 \) Using Point \( P(5, 9) \)
To verify that \( a = 2 \), we substitute the point \( (5, 9) \) into the equation:
\[
9 = 2(5 - 3)^2 + 1
\]
\[
9 = 2(2)^2 + 1
\]
\[
9 = 2 \cdot 4 + 1
\]
\[
9 = 8 + 1
\]
\[
9 = 9
\]
This confirms that our value of \( a = 2 \) is correct.

### Final Answers:
**a.** The roller coaster journey starts at a height of **19 meters**.

**b.** The values of \( h \) and \( k \) are **3** and **1**, respectively.

**c.** The equation of the curve is:
\[
H = 2(x - 3)^2 + 1
\]

**d.** Using the equation, we verified that \( a = 2 \).

Would you like more details or have any questions?

### Related Questions:
1. How would the equation change if the vertex was at a different point?
2. What does the value of \( a \) indicate about the shape of the parabola?
3. How can you find the vertex form of a parabola if only given three points?
4. What happens to the graph if the value of \( k \) changes?
5. Can you find the roots of the equation \( H = 0 \)?

### Tip:
Remember that the vertex form of a quadratic equation, \( a(x - h)^2 + k \), makes it easier to determine the maximum or minimum point of the graph directly.

The diagram shows part of a roller coaster. The graph of the curve H = a(x - h)^2 + k, where H is the height in meters and x is the distance from the starting point for 0 ≤ x ≤ 6. The y-intercept is (0, 19). The vertex is at the point (3, 1), and the point P(5, 9) is on the graph. Solve the following:

a) At what height do you begin your roller coaster journey?
b) Write down the value of h and k.
c) Write down the equation of the curve.
d) Using your answer in part c, show that a = 2.

Solve a quadratic equation to determine the shape of a roller coaster curve, including finding the vertex, equation of the curve, and determining constants. Suitable for students in Grades 9-11.

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