The question asks you to use the graph of the function $f$, plotted with a solid line, and sketch the graph of $g(x) = f(x+2) - 2$.

Let's break down what $g(x) = f(x+2) - 2$ means in terms of transformations of the graph of $f(x)$:

1. $f(x+2)$: This represents a horizontal shift. The graph shifts 2 units to the left (because adding inside the function causes a shift in the opposite direction).
2. $-2$: This represents a vertical shift. The graph shifts 2 units downward.

So, to obtain the graph of $g(x)$ from the graph of $f(x)$, we shift the graph of $f(x)$ 2 units to the left and 2 units downward.

Now, let's analyze the options:

• Option A: This shows a graph shifted 2 units to the right, which is incorrect.
• Option B: This shows a graph shifted 2 units to the left, but no vertical shift.
• Option C: This graph is shifted 2 units to the left and 2 units downward, which is correct.
• Option D: This shows a graph shifted downward but without the horizontal shift.

Thus, the correct answer is Option C.

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

Here are 5 related questions to explore further:

1. What happens if the function $g(x)$ is $f(x-3) + 4$?
2. How would the transformation change if the equation was $g(x) = 2f(x)$?
3. What are the effects of a vertical stretch on a graph?
4. Can you explain how a reflection across the x-axis affects the graph of $f(x)$?
5. How do you find the vertex of a transformed quadratic function?

Tip:

For any transformation, changes inside the function (like $x+2$) affect the graph horizontally, and changes outside the function (like $-2$) affect it vertically.