The given transformation is $y + 1 = f\left( \frac{1}{4}x - 1 \right)$.

Step-by-Step Breakdown:

1. Horizontal Transformation:

   • $\frac{1}{4}x - 1$ represents two transformations:
     • Horizontal stretch by a factor of 4. To account for this, multiply the $x$-coordinate by 4.
     • Horizontal shift to the right by 1. To account for this, add 1 to the $x$-coordinate.

   Applying this to the point $(-3, 8)$: $x' = 4 \times (-3 + 1) = 4 \times (-2) = -8$

2. Vertical Transformation:

   • $y + 1 = f(\text{...})$ indicates a vertical shift down by 1. So, subtract 1 from the $y$-coordinate.

   Applying this to the point $(-3, 8)$: $y' = 8 - 1 = 7$

Final Coordinates:

After the transformation, the new coordinates of the point are $(-8, 7)$.

Would you like more details on the transformations, or do you have any questions?

Here are 5 related questions to explore:

1. What happens to the point if the transformation includes a reflection across the y-axis?
2. How would the transformation affect the slope of a line passing through the original point?
3. What is the effect of applying only the horizontal transformation to the point?
4. How do you reverse a vertical shift in a transformation equation?
5. What happens if we change the stretch factor from 4 to another number?

Tip: Always handle horizontal transformations (stretch/compression and shifts) before applying vertical changes!