What kind of transformation converts the graph of f(x)=5|x+5|–8 into the graph of g(x)=5|x–5|–8?

The transformation that converts the graph of \( f(x) = 5|x + 5| - 8 \) into the graph of \( g(x) = 5|x - 5| - 8 \) is a **horizontal translation**.

Specifically, this is a shift of the graph of \( f(x) \) **10 units to the right** to obtain the graph of \( g(x) \).

### Explanation:
1. The function \( f(x) = 5|x + 5| - 8 \) has a vertex at \( (-5, -8) \).
2. The function \( g(x) = 5|x - 5| - 8 \) has a vertex at \( (5, -8) \).
3. To transform \( f(x) \) into \( g(x) \), you need to move the graph from the vertex at \( (-5, -8) \) to the vertex at \( (5, -8) \).
4. This movement is achieved by shifting the graph **10 units to the right**.

The other components of the function (the coefficient of 5 and the constant term -8) remain unchanged, indicating that there is no vertical shift, stretch, or compression involved—just the horizontal translation.

Would you like a detailed graphical representation or have any questions? 

Here are some related questions to explore:

1. What effect does changing the coefficient of \(|x|\) have on the graph of an absolute value function?
2. How would you translate \( f(x) = 5|x + 5| - 8 \) vertically to match \( g(x) = 5|x - 5| - 3 \)?
3. What happens to the graph of an absolute value function when you reflect it over the y-axis?
4. How do you find the vertex of a function in the form \( f(x) = a|x - h| + k \)?
5. How does shifting the graph horizontally affect the domain and range of an absolute value function?

**Tip:** When analyzing transformations, always identify changes to the \( x \)- and \( y \)-values separately to determine horizontal and vertical shifts, reflections, stretches, and compressions.

Discover how to apply horizontal translations to transform the graph of f(x) = 5|x + 5| – 8 into g(x) = 5|x – 5| – 8. This detailed explanation covers absolute value functions and graph transformations, ideal for students in Grades 9-11.

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