What kind of transformation converts the graph of 

f

(

x

)

=

–

5

|

x

–

7

|

–

7

 into the graph of 

g

(

x

)

=

–

5

|

x

+

1

|

–

7

?

To convert the graph of \( f(x) = -5|x - 7| - 7 \) into the graph of \( g(x) = -5|x + 1| - 7 \), you apply a **horizontal translation**.

### Transformation Details:
- The coefficient of the absolute value term and the constant term outside the absolute value (i.e., \(-7\)) are the same in both functions, so the vertical position and shape of the graph remain unchanged.
- The difference lies in the expression inside the absolute value: \( |x - 7| \) in \( f(x) \) and \( |x + 1| \) in \( g(x) \).

**Horizontal Translation**:
- \( f(x) \) has \( x - 7 \) inside the absolute value, which means the graph of \( f(x) \) is shifted **7 units to the right** from the standard \( |x| \) graph.
- \( g(x) \) has \( x + 1 \), equivalent to \( x - (-1) \), meaning the graph is shifted **1 unit to the left** from the standard \( |x| \) graph.

To move from \( x - 7 \) to \( x + 1 \), you translate the graph **8 units to the left**.

Would you like more details or have any questions?

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**Related Questions**:
1. What is the impact of shifting the graph horizontally on the vertex of the absolute value function?
2. How would you describe the transformation if the graph of \( f(x) \) were shifted 3 units to the right instead?
3. How does a vertical translation compare to a horizontal translation in terms of graph movement?
4. What would be the equation if the graph was shifted 8 units to the right instead?
5. How does changing the coefficient of the absolute value affect the steepness of the graph?

**Tip**: Horizontal translations inside the absolute value function can be identified by examining the changes in the terms inside the absolute value.

Understand how to transform absolute value functions with a horizontal translation. Learn through an example of converting f(x) = -5|x - 7| - 7 to g(x) = -5|x + 1| - 7, including detailed explanations and step-by-step process. Suitable for Grades 11-12.

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