To find the point that is symmetrical to the point \((-1, -3)\) with respect to the graph of the function \(y = x^2 + 12x + 8\), we first need to identify the vertex of the parabola described by the function.
The given quadratic function can be expressed in standard form as follows:
- Complete the square or use the vertex formula \(x = -\frac{b}{2a}\).
The function is \(y = x^2 + 12x + 8\), where \(a = 1\) and \(b = 12\).
- Calculate the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} = -\frac{12}{2(1)} = -6. \]
- Substitute \(x = -6\) back into the equation to find the y-coordinate of the vertex:
\[ y = (-6)^2 + 12(-6) + 8 = 36 - 72 + 8 = -28. \]
So the vertex of the parabola is \((-6, -28)\).
Now, to find the reflected point, we need to calculate the distance from the point \((-1, -3)\) to the vertex \((-6, -28)\):
- The x-coordinate distance:
\[ \text{Distance in x} = -1 - (-6) = -1 + 6 = 5. \]
- The y-coordinate distance:
\[ \text{Distance in y} = -3 - (-28) = -3 + 28 = 25. \]
Now, to reflect the point \((-1, -3)\) across the vertex \((-6, -28)\), we'll move the same distance from the vertex in the opposite direction:
- For the x-coordinate:
\[ x' = -6 - 5 = -11. \]
- For the y-coordinate:
\[ y' = -28 - 25 = -53. \]
Thus, the point that is symmetrical to \((-1, -3)\) with respect to the graph is:
\[ \boxed{(-11, -53)}. \]