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The problem requires converting a quadratic equation into vertex form by completing the square, and then using that form to analyze the grappling hook's flight.

### Given Equation:
\[ y = -9.8t^2 + 11.7t - 3.828 \]

### Steps to Complete the Square:

1. **Factor out the coefficient of \( t^2 \) from the first two terms:**
   \[
   y = -9.8\left(t^2 - \frac{11.7}{9.8}t\right) - 3.828
   \]
   Simplify the coefficient:
   \[
   y = -9.8\left(t^2 - 1.194t\right) - 3.828
   \]

2. **Complete the square inside the parentheses:**
   - Take half of 1.194, square it:
     \[
     \left(\frac{1.194}{2}\right)^2 = 0.357
     \]
   - Add and subtract 0.357 inside the parentheses:
     \[
     y = -9.8\left(t^2 - 1.194t + 0.357 - 0.357\right) - 3.828
     \]
   - Rearrange:
     \[
     y = -9.8\left((t - 0.597)^2 - 0.357\right) - 3.828
     \]
   - Distribute \(-9.8\) and simplify:
     \[
     y = -9.8(t - 0.597)^2 + 3.5 - 3.828
     \]
     \[
     y = -9.8(t - 0.597)^2 - 0.328
     \]

### Vertex Form:
The vertex form is:
\[
y = -9.8(t - 0.597)^2 - 0.328
\]

### Analysis:

1. **Vertex Coordinates:**
   - The vertex is at \( t = 0.597 \), \( y = -0.328 \).
   - Since the coefficient of \((t - 0.597)^2\) is negative, the parabola opens downwards.

2. **Physical Interpretation:**
   - The vertex represents the maximum height of the grappling hook. 
   - Given that the maximum height (\( y = -0.328 \)) is negative, this implies that the hook does not reach the top of the cliff (12 feet above).

### Answers to Dropdown Questions:

1. **Vertex Form:**
   - \( y = -9.8(t - 0.597)^2 - 0.328 \)

2. **Vertex Location and Parabola Direction:**
   - Vertex at \( (0.597, -0.328) \), Parabola opens downward.

3. **Grappling Hook's Behavior:**
   - The grappling hook will not reach the top of the cliff.

4. **Solutions to the Equation:**
   - Since the hook's path does not reach a height of 12 feet, the solutions related to reaching that height are not real or are not valid in the context of this problem.

Would you like more details on any specific part of the problem, or do you have any further questions?

### Related Questions:
1. How would the vertex change if the initial velocity was higher?
2. What is the significance of the discriminant in this context?
3. How could you determine the time it takes for the grappling hook to reach its maximum height?
4. What does the negative value of the vertex indicate about the grappling hook's motion?
5. How would the equation change if the cliff height was different?

### Tip:
When completing the square, always remember to balance the equation by adjusting both sides accordingly if needed.

Allison is practicing her survival skills while out hiking. She reaches the bottom of a cliff and decides that to continue on she must get to the top, which is more than 12 feet above her. She decides to use her rope and grappling hook to help her scale the cliff. The flight of the grappling hook can be modeled with the equation: y = -9.8t^2 + 11.7t - 3.828 where t is the time in seconds and y is the height from the top of the cliff. By completing the square, you can determine how long it will take for the hook to reach the top of the cliff, modeled by the x-axis.

Explore how to solve a quadratic equation modeling the flight of a grappling hook using the method of completing the square. Understand the key concepts of algebra, including the vertex form of a quadratic equation and its application to real-world scenarios. Suitable for students in Grades 10-12.

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