What kind of transformation converts the graph of f(x)= –7|x–2|+8 into the graph of g(x)=–7|x+7|+8?


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.

What kind of transformation converts the graph of f(x) = –7|x–2| + 8 into the graph of g(x) = –7|x+7| + 8?

Explore how to transform the graph of f(x) = –7|x–2| + 8 into g(x) = –7|x+7| + 8 using horizontal translation. This solution breaks down key concepts in algebra and absolute value function transformations, suitable for high school students in Grades 9-12.

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