To determine the transformation that converts the graph of $f(x) = -10 |x + 9| + 5$ into the graph of $g(x) = -10 |x + 8| + 1$, we need to compare the two equations and analyze the changes between them.

Step-by-step analysis:

1. Horizontal Translation:

   • In $f(x)$, the term inside the absolute value is $x + 9$, while in $g(x)$, the term inside the absolute value is $x + 8$.
   • This indicates a horizontal shift. The change from $x + 9$ to $x + 8$ means the graph shifts 1 unit to the right because decreasing the number inside the absolute value shifts the graph in the negative direction (right).

2. Vertical Translation:

   • In $f(x)$, the constant term outside the absolute value is $+5$, while in $g(x)$, the constant term outside the absolute value is $+1$.
   • This indicates a vertical shift. The change from $+5$ to $+1$ means the graph shifts 4 units downward.

Final Transformation:

• Horizontal shift 1 unit to the right.
• Vertical shift 4 units downward.

Would you like further clarification or additional details on this transformation?

Here are five related questions to consider:

1. How would the graph change if we shifted $f(x)$ by 2 units to the left?
2. What happens to the graph of an absolute value function if you change the coefficient inside the absolute value?
3. How does the reflection of an absolute value function work?
4. What is the general form of transformations involving absolute value functions?
5. How would the transformation differ if the coefficient of $x$ inside the absolute value was changed?

Tip: Horizontal shifts are always opposite in direction to the sign of the number added/subtracted inside the absolute value function.