We are given that the point $(8, -5)$ lies on the graph of $f(x)$. We are also given the transformation $g(x) = 2f(x-4) + 6$. We need to find the corresponding point on the graph of $g(x)$.

1. Problem Description

We are given that the point $(8, -5)$ lies on the graph of $f(x)$. We are also given the transformation $g(x) = 2f(x-4) + 6$. We need to find the corresponding point on the graph of $g(x)$.

2. Solution Steps

Let $(x, y)$ be a point on the graph of $f(x)$. This means $y = f(x)$. We are given that $(8, -5)$ is on the graph of $f(x)$, so $f(8) = -5$.

Now consider the function $g(x) = 2f(x-4) + 6$. Let $(a, b)$ be a point on the graph of $g(x)$, which means $b = g(a)$. Substituting the expression for $g(x)$, we have $b = 2f(a-4) + 6$.

We want to relate this to the point $(8, -5)$ on the graph of $f(x)$. We have $f(8) = -5$. We want to find a value of $a$ such that $a-4 = 8$. Solving for $a$, we get $a = 8+4 = 12$.

Substituting $a=12$ into the equation for $b$, we have

$b = 2f(12-4) + 6 = 2f(8) + 6 = 2(-5) + 6 = -10 + 6 = -4$.

So the corresponding point on the graph of $g(x)$ is $(12, -4)$.

3. Final Answer

(12, -4)