Let's analyze the image you provided.

From what I can see:

1. The graph shows two functions:

   • The red line labeled $g(x)$
   • The blue line labeled $f(x) = |x|$, which is the standard absolute value function.

2. The goal is to find the formula for $g(x)$.

Observations:

• The blue graph is $f(x) = |x|$, which represents the V-shaped absolute value function.
• The red graph appears to be a transformation of the blue one. Specifically:
  • It has the same shape as $|x|$, but it is translated vertically.

Finding the formula for $g(x)$:

• The red graph seems to be a vertical shift of $f(x) = |x|$.
• Notice that the vertex of the red graph is at $(0, 8)$ (instead of at $(0, 0)$ for $f(x) = |x|$).
• This suggests a vertical shift of 8 units upward.

Therefore, the formula for $g(x)$ is: $g(x) = |x| + 8$

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. How do vertical shifts affect the equation of absolute value functions?
2. What changes would result in horizontal shifts for $f(x) = |x|$?
3. How would the equation of $f(x) = |x|$ change if it were reflected across the x-axis?
4. Can you explain how scaling (compressing/stretching) affects the absolute value function?
5. What is the general form of transformations for the function $f(x) = |x|$?

Tip: When determining shifts and transformations of a graph, always compare the vertex and slope of key segments.