To understand the transformation from $f(x)$ to $g(x)$, let's first analyze the graph:

1. Function $f(x)$: The function $f(x)$ is a piecewise linear function, forming a "V" shape. This indicates that $f(x)$ is most likely the absolute value function: $f(x) = |x|$ This fits the graph of $f(x)$, where the point of the "V" occurs at $x = 0$, and the slopes of the two lines forming the "V" are $1$ and $-1$.

2. Function $g(x)$: The function $g(x)$ appears to be a transformation of $f(x)$, as it looks like it involves a shift and a reflection.

Step 1: Reflection

• $g(x)$ seems to be reflected over the x-axis relative to $f(x)$. This is a vertical reflection, which changes $f(x) = |x|$ to $-f(x)$, meaning: $g(x) = -|x|$

Step 2: Shift

• $g(x)$ is shifted horizontally to the right by 4 units. To account for this, we replace $x$ with $(x - 4)$ in the function: $g(x) = -|x - 4|$

Step 3: Vertical Shift

• Finally, $g(x)$ is shifted vertically down by 3 units. To incorporate this, we subtract 3 from the function: $g(x) = -|x - 4| - 3$

Conclusion:

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

1. A vertical reflection,
2. A horizontal shift to the right by 4 units,
3. A vertical shift down by 3 units.

The equation for $g(x)$ in terms of $f(x) = x$ (absolute value function) is: $g(x) = -|x - 4| - 3$

Would you like more details on transformations or have any further questions?

Here are five related questions for practice:

1. What happens to a graph when you apply a vertical stretch or compression?
2. How would you write the equation of $g(x)$ if it were shifted left instead of right?
3. What is the vertex of the function $g(x)$?
4. How does reflecting a graph over the y-axis change the equation?
5. What other types of transformations can be applied to a function?

Tip: Always track the effect of each transformation (shift, reflection, stretch/compression) one step at a time to avoid confusion.