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Find X and y coordinate of the vertex and value of a?

To find the vertex coordinates of the quadratic function \( q(x) = -(x - 7)^2 - 81 \) and the value of \( a \) for the polynomial \( p(x) = a(x - 1)(x - 5)(x - 8)(x - 10) \) passing through this vertex, we need to follow these steps:

### Step 1: Find the Vertex of the Quadratic Function
The quadratic function given is:
\[ q(x) = -(x - 7)^2 - 81 \]

The vertex form of a quadratic function \( q(x) = a(x - h)^2 + k \) shows that the vertex is at \((h, k)\). Here:
- \( a = -1 \)
- \( h = 7 \)
- \( k = -81 \)

Thus, the vertex is at \((7, -81)\).

### Step 2: Determine the Value of \( a \)
The polynomial \( p(x) = a(x - 1)(x - 5)(x - 8)(x - 10) \) passes through the vertex \((7, -81)\). Therefore, we can substitute \( x = 7 \) and \( y = -81 \) into \( p(x) \) to find \( a \).

\[ p(7) = -81 \]

\[ -81 = a(7 - 1)(7 - 5)(7 - 8)(7 - 10) \]
\[ -81 = a(6)(2)(-1)(-3) \]
\[ -81 = a(6 \cdot 2 \cdot -1 \cdot -3) \]
\[ -81 = a(36) \]

Solving for \( a \):

\[ a = \frac{-81}{36} \]
\[ a = -\frac{9}{4} \]

### Summary
- The coordinates of the vertex are \((7, -81)\).
- The value of \( a \) is \(-\frac{9}{4}\).

Would you like more details on any of these steps or have any other questions?

### Related Questions
1. How do you convert a quadratic function from standard form to vertex form?
2. What are the steps to find the roots of a polynomial?
3. Can you explain the properties of the vertex form of a quadratic equation?
4. How do you determine the axis of symmetry for a quadratic function?
5. What is the significance of the leading coefficient in a polynomial function?
6. How do you find the y-intercept of a polynomial function?
7. What methods can be used to factor a polynomial?
8. How do polynomial functions differ from quadratic functions in their graphs and properties?

### Tip
When finding the vertex of a quadratic function, remember that the vertex form \( a(x - h)^2 + k \) directly provides the vertex coordinates \((h, k)\), making it easier than converting from standard form.

Learn how to find the vertex coordinates of a quadratic function and determine the value of a in a polynomial function with a detailed step-by-step solution. This problem covers key concepts including the vertex form of a quadratic function and polynomial evaluation. Suitable for Grades 11-12.

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