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.