a = -1, x coordinate vertex is 7 and y coordinate vertex is -9

Let's clarify the steps to correct any misunderstanding.

### Correcting the Vertex Coordinates and Value of \(a\)

You mentioned that \(a = -1\), the \(x\)-coordinate of the vertex is 7, and the \(y\)-coordinate of the vertex is -9. This suggests we should review the vertex of the quadratic function and the polynomial \(p(x)\).

### Step 1: Re-evaluating the Vertex of the Quadratic Function
The given quadratic function is:
\[ q(x) = -(x - 7)^2 - 9 \]

In this form:
- The coefficient \(a = -1\)
- The vertex \((h, k)\) has \(h = 7\) and \(k = -9\)

Thus, the vertex is indeed at \((7, -9)\).

### Step 2: Correcting the Value of \(a\)
The polynomial \( p(x) = a(x - 1)(x - 5)(x - 8)(x - 10) \) passes through the vertex \((7, -9)\). We will substitute \( x = 7 \) and \( y = -9 \) to find \( a \).

\[ p(7) = -9 \]

Substituting \( x = 7 \):

\[ -9 = a(7 - 1)(7 - 5)(7 - 8)(7 - 10) \]
\[ -9 = a(6)(2)(-1)(-3) \]
\[ -9 = a(6 \cdot 2 \cdot -1 \cdot -3) \]
\[ -9 = a(36) \]

Solving for \( a \):

\[ a = \frac{-9}{36} \]
\[ a = -\frac{1}{4} \]

### Summary
- The coordinates of the vertex are \((7, -9)\).
- The value of \( a \) is \(-\frac{1}{4}\).

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

### Related Questions
1. How do you verify the vertex of a quadratic function?
2. What are the steps to find the roots of a polynomial function?
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 determining the value of a coefficient in a polynomial, always substitute known values of \(x\) and \(y\) into the polynomial equation to solve for the unknown coefficient accurately.

Explore the vertex of a quadratic function and determine the coefficient using specific examples and detailed calculations. Learn how to solve for the value of 'a' in polynomial functions through step-by-step guidance. Suitable for Grades 10-12.

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