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.