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The graph provided shows a transformation of the function \( f(x) = |x| \). The general form for a translated absolute value function is:

\[
g(x) = a |x - h| + k
\]

where:
- \( a \) represents the vertical stretch or compression (and possibly reflection),
- \( h \) represents the horizontal shift,
- \( k \) represents the vertical shift.

### Step-by-Step Solution:

1. **Vertex (h, k):**
   The vertex of the basic function \( f(x) = |x| \) is at \( (0, 0) \). From the graph, the vertex of \( g(x) \) is at \( (0, 4) \). This means \( h = 0 \) and \( k = 4 \).

2. **Determining the Slope (a):**
   For the basic function \( f(x) = |x| \), the slope of the right branch (where \( x \geq 0 \)) is 1, and the slope of the left branch (where \( x \leq 0 \)) is -1. For \( g(x) \), the slope on the right is 2 (since the function rises 2 units for every 1 unit to the right) and -2 on the left.

   Therefore, \( a = 2 \).

### Final Function:

\[
g(x) = 2|x| + 4
\]

This is the equation for the translated function \( g(x) \).

Would you like more details or have any other questions?

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Here are 5 questions related to this concept:

1. What would the graph look like if the value of \( a \) was negative?
2. How does the value of \( k \) affect the vertical position of the graph?
3. What changes if \( h \) is not zero in the equation \( g(x) = a|x - h| + k \)?
4. How would you find the equation of a graph if it's compressed horizontally?
5. Can you describe how to shift the graph of \( f(x) = |x| \) both vertically and horizontally?

**Tip:** Remember, the parameter \( a \) controls the steepness of the graph. A larger \( |a| \) value stretches the graph vertically, while a smaller \( |a| \) value compresses it.

The graph shows g(x), which is a translation of f(x) = |x|. Write the function rule for g(x).

Explore how to find the equation of the translated absolute value function g(x) from f(x) = |x|. The step-by-step solution includes identifying vertical and horizontal shifts as well as slope changes. Suitable for students in Grades 9-11.

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