To analyze the transformation of $f(x)$ to $g(x)$ from the provided graph, we need to compare the two functions visually.

Step-by-Step Analysis:

1. $f(x)$ is a V-shaped function, suggesting it is an absolute value function. From the graph, $f(x)$ seems to be $f(x) = |x|$, which creates the characteristic "V" shape.

2. $g(x)$ appears to be a transformation of $f(x)$, specifically a reflection and a translation:

   • It is reflected across the x-axis (since the slope of $g(x)$ is downward).
   • There is a vertical translation downwards, and the function seems to be shifted down by 6 units.

General Transformation:

The transformation from $f(x)$ to $g(x)$ involves:

• A reflection across the x-axis.
• A vertical translation downward by 6 units.

Thus, the equation for $g(x)$ in terms of $f(x)$ is: $g(x) = -f(x) - 6$ Since $f(x) = |x|$, the equation of $g(x)$ becomes: $g(x) = -|x| - 6$

This means that:

• The reflection is represented by the negative sign.
• The shift downward by 6 units is captured by subtracting 6.

Summary:

• The function $f(x) = |x|$ has been reflected over the x-axis and translated downward by 6 units to form $g(x)$.
• The equation for $g(x)$ in terms of $f(x) = |x|$ is $g(x) = -|x| - 6$.

Do you need any further details or have any questions?

Here are 5 related questions to further explore:

1. What is the general form of a vertical translation and reflection in function transformations?
2. How would the graph change if $g(x)$ had been shifted upward instead of downward?
3. What happens to the function if $f(x) = |x|$ is shifted horizontally?
4. How can you derive the vertex of an absolute value function from its equation?
5. How would you reflect $f(x)$ over the y-axis instead of the x-axis?

Tip: When analyzing transformations, always look for reflections, translations, stretches, and compressions to see how they affect the graph visually.