The problem you provided contains a graph of the function $f(x)$ and has several questions related to it. Here's a breakdown of the problem and its questions:

Graph Interpretation:

The function $f(x)$ appears to be a piecewise linear function, with the graph forming peaks and valleys. Here are some key points visible from the graph:

1. The function has a peak at $x = -2$ where $f(-2) = 3$.
2. It decreases until $x = 0$, where $f(0) = -2$.
3. Then it increases to a peak at $x = 2$, where $f(2) = 3$.
4. Finally, it decreases again beyond $x = 2$.

Now, let’s address each question.

(a) Determine the value of $f(-2)$:

From the graph, we can see that at $x = -2$, the function reaches its peak, so: $f(-2) = 3$

(b) If $g(x) = f(x) + 2$, determine the value of $g(0)$:

We know that $f(0) = -2$ from the graph. Using the given function $g(x) = f(x) + 2$, we substitute $x = 0$: $g(0) = f(0) + 2 = -2 + 2 = 0$

(c) Sketch the graph of $g(x)$:

To sketch the graph of $g(x) = f(x) + 2$, we shift the entire graph of $f(x)$ upward by 2 units. This means:

• The peak at $f(-2) = 3$ will now be at $g(-2) = 5$.
• The valley at $f(0) = -2$ will move to $g(0) = 0$.
• The peak at $f(2) = 3$ will shift to $g(2) = 5$.

So, the graph of $g(x)$ will have the same shape as $f(x)$, just shifted vertically by 2 units.

Would you like more details on any of the answers or further clarification?

Here are 5 questions for further exploration:

1. How would you determine the slope of each segment in the piecewise function $f(x)$?
2. What changes to the graph would occur if $g(x) = f(x) - 3$?
3. Can you write the piecewise linear equations for each segment of $f(x)$?
4. How does the vertical shift affect the zeros of the function $g(x)$?
5. Could you find the domain and range of both $f(x)$ and $g(x)$?

Tip: When shifting a graph vertically by $k$, each point $(x, y)$ on the graph moves to $(x, y+k)$.