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?

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.