The transformation that converts the graph of $f(x) = -7|x - 2| + 8$ into the graph of $g(x) = -7|x + 7| + 8$ is a horizontal translation to the left by 9 units.

Here’s how we deduce that:

1. The general form of the given function is $f(x) = a|x - h| + k$, where:

   • $a$ controls the vertical stretch/compression and reflection,
   • $h$ controls the horizontal translation,
   • $k$ controls the vertical translation.

2. In $f(x) = -7|x - 2| + 8$, the graph is translated 2 units to the right due to $(x - 2)$.

3. In $g(x) = -7|x + 7| + 8$, the graph is translated 7 units to the left due to $(x + 7)$.

4. To go from $f(x)$ to $g(x)$, the horizontal shift changes from $2$ to $-7$, meaning the graph has been shifted 9 units to the left (because $-7 - 2 = -9$).

Final Transformation:

• Horizontal translation 9 units left.

Would you like a more detailed explanation or have any questions?

Here are five related questions to consider:

1. How do vertical translations affect absolute value functions?
2. What happens to the graph if you change the coefficient $-7$ to a different value?
3. How would the transformation change if the equation included a vertical shift?
4. Can you describe the effect of reflecting the graph across the x-axis?
5. What kind of transformation would move the graph to the right instead of the left?

Tip: Horizontal transformations inside the absolute value function act in the opposite direction of their sign.