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.