The point $(1, -12)$ is on the graph of $f(x)$. Given the transformation $g(x) = \frac{1}{3}f(x+6) - 3$, we need to find the corresponding point on the graph of $g(x)$.

1. Problem Description

The point $(1, -12)$ is on the graph of $f(x)$. Given the transformation $g(x) = \frac{1}{3}f(x+6) - 3$, we need to find the corresponding point on the graph of $g(x)$.

2. Solution Steps

Let $(x', y')$ be the corresponding point on the graph of $g(x)$.

We are given $g(x) = \frac{1}{3}f(x+6) - 3$. This transformation involves a horizontal shift, a vertical stretch/compression, and a vertical shift.

The argument $x+6$ inside the function $f$ indicates a horizontal shift to the left by 6 units. Thus, $x' + 6 = 1$, which means $x' = 1 - 6 = -5$.

The coefficient $\frac{1}{3}$ in front of $f(x+6)$ indicates a vertical compression by a factor of

3. The term $-3$ indicates a vertical shift downward by 3 units. Therefore, $y' = \frac{1}{3}f(x'+6) - 3 = \frac{1}{3}f(-5+6) - 3 = \frac{1}{3}f(1) - 3$.

Since $(1, -12)$ is on the graph of $f(x)$, we have $f(1) = -12$.

Therefore, $y' = \frac{1}{3}(-12) - 3 = -4 - 3 = -7$.

So the corresponding point on the graph of $g(x)$ is $(-5, -7)$.

3. Final Answer

$(-5, -7)$