The graph you provided appears to represent a function with a "V" shape, similar to an absolute value function, $f(x)$, with three key points at $(-4, 0)$, $(4, 0)$, and $(0, -8)$.

To sketch the graph of $y = |f(x)|$, we make the following observations:

1. Positive Values of $f(x)$: Any points where $f(x) \geq 0$ (i.e., points above or on the x-axis) will remain unchanged since the absolute value of a non-negative number is itself.

2. Negative Values of $f(x)$: Any points where $f(x) < 0$ (i.e., points below the x-axis) will reflect across the x-axis, turning those values positive. This means that any parts of the graph below the x-axis will be mirrored above the x-axis.

How to sketch $y = |f(x)|$:

• For the points on the graph that lie on or above the x-axis (like at $(-4, 0)$, $(4, 0)$), those points remain the same.
• For the point at $(0, -8)$, it will reflect across the x-axis to become $(0, 8)$.

Thus, the portion of the graph below the x-axis will be "flipped" above it, while the portion already above or on the x-axis will remain unchanged.

Would you like a detailed sketch of the modified graph, or do you have any other questions?

Here are 5 related questions for further understanding:

1. What is the domain and range of the original function $f(x)$ and its transformation $|f(x)|$?
2. How do we find the zeros of $|f(x)|$?
3. Can this method be applied to any type of function? Why or why not?
4. What other transformations can be performed on $f(x)$ besides taking the absolute value?
5. How does the vertex of the function change when transforming $f(x)$ to $|f(x)|$?

Tip: When working with absolute value transformations, always pay close attention to how negative values of the function change, as they reflect symmetrically across the x-axis.