The problem is asking you to graph the function $g(x) = f(x) + 3$ based on the given graph of $y = f(x)$.

Key Concept:

The transformation $g(x) = f(x) + 3$ represents a vertical shift of the function $f(x)$ upwards by 3 units. This means that each point on the graph of $f(x)$ will be moved 3 units higher on the $y$-axis.

Procedure:

1. Identify key points on the graph of $f(x)$. For example, look at the points where the graph changes direction or passes through the axes.
2. Shift each of these points 3 units upwards. That means, for any point $(x, y)$ on $f(x)$, the corresponding point on $g(x)$ will be $(x, y+3)$.
3. The new graph should maintain the same shape as the original, but just shifted vertically.

Comparison with the Options:

From the given options:

• Option A shows a graph shifted downwards, which would correspond to a subtraction transformation, not an addition.
• Option B shows the graph at the same vertical level, meaning no vertical shift, which is incorrect.
• Option C shows the graph shifted correctly upwards by 3 units, maintaining the shape of the original graph.
• Option D shows an incorrect vertical shift.

Thus, the correct answer is Option C.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would the graph change if the transformation were $g(x) = f(x) - 2$?
2. What is the effect of the transformation $g(x) = 2f(x)$ on the graph?
3. How do horizontal shifts differ from vertical shifts in graph transformations?
4. Can you describe the transformation represented by $g(x) = f(x - 2)$?
5. How would the graph of $f(x)$ change under a reflection over the x-axis?

Tip: Vertical shifts only affect the $y$-values of the points, not the $x$-values.