The problem gives us the vertex of a quadratic function and another point that the function passes through. We are asked to find the values of $a$, $h$, and $k$ in the vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$. The vertex is given as $(-9, 7)$, and the other point is $(-5, 6)$.

1. Problem Description

The problem gives us the vertex of a quadratic function and another point that the function passes through. We are asked to find the values of $a$, $h$, and $k$ in the vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$. The vertex is given as $(-9, 7)$, and the other point is $(-5, 6)$.

2. Solution Steps

The vertex form of a quadratic function is given by:

$f(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola. In this problem, the vertex is $(-9, 7)$, so $h = -9$ and $k = 7$.

Therefore, we can rewrite the equation as:

$f(x) = a(x - (-9))^2 + 7$

$f(x) = a(x + 9)^2 + 7$

We are also given that the function passes through the point $(-5, 6)$. This means that when $x = -5$, $f(x) = 6$. We can substitute these values into the equation to solve for $a$:

$6 = a(-5 + 9)^2 + 7$

$6 = a(4)^2 + 7$

$6 = 16a + 7$

Subtract 7 from both sides:

$6 - 7 = 16a$

$-1 = 16a$

Divide by 16:

$a = -\frac{1}{16}$

So, $a = -\frac{1}{16}$, $h = -9$, and $k = 7$.

3. Final Answer

$a = -\frac{1}{16}$

$h = -9$

$k = 7$