We are given a quadratic function in vertex form, $f(x) = a(x - h)^2 + k$. The vertex of the quadratic is $(-10, 7)$, and the function passes through the point $(5, 8)$. We need to find the values of $a$, $h$, and $k$.

1. Problem Description

We are given a quadratic function in vertex form, $f(x) = a(x - h)^2 + k$. The vertex of the quadratic is $(-10, 7)$, and the function passes through the point $(5, 8)$. We need to find the values of $a$, $h$, and $k$.

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.

We are given that the vertex is $(-10, 7)$, so $h = -10$ and $k = 7$.

Substituting these values into the vertex form, we have

$f(x) = a(x - (-10))^2 + 7$

$f(x) = a(x + 10)^2 + 7$

Now, we are given that the function passes through the point $(5, 8)$, so we can substitute $x = 5$ and $f(x) = 8$ into the equation:

$8 = a(5 + 10)^2 + 7$

$8 = a(15)^2 + 7$

$8 = 225a + 7$

Subtracting 7 from both sides, we get

$1 = 225a$

Dividing by 225, we find

$a = \frac{1}{225}$

Therefore, $a = \frac{1}{225}$, $h = -10$, and $k = 7$.

3. Final Answer

$a = \frac{1}{225}$

$h = -10$

$k = 7$