The task requires graphing the function $g(x) = f(x) + 3$ based on the provided graph of $f(x)$, which is shown in red on the coordinate grid. Let's break down how to achieve this:

1. Understanding the Transformation: The function $g(x) = f(x) + 3$ represents a vertical translation of the graph of $f(x)$. Specifically, every point on the graph of $f(x)$ will be shifted upward by 3 units.

2. Graphing $g(x)$:

   • Locate each point on the graph of $f(x)$. Let's call these points $(x_1, y_1)$.
   • For each point $(x_1, y_1)$, the corresponding point on the graph of $g(x)$ will be $(x_1, y_1 + 3)$.
   • Move each point of the graph of $f(x)$ vertically upward by 3 units and plot the new points.
   • Connect these new points to create the graph of $g(x)$.

3. Visualizing the Graph:

   • For the provided red graph, identify the key points where the lines change direction.
   • Shift these key points up by 3 units.
   • Redraw the lines maintaining the same shape as the original graph but shifted upward.

Would you like more details on this transformation process or how the new graph would look?

5 Relative Questions:

1. What would be the effect of the transformation $g(x) = f(x) - 2$?
2. How would you graph $g(x) = 2f(x)$ based on the original graph of $f(x)$?
3. What happens if the transformation is $g(x) = f(x - 3)$ instead?
4. How do horizontal translations differ from vertical translations?
5. Can you describe how to reflect the graph of $f(x)$ across the x-axis?

Tip:

When applying transformations, always remember that changes outside the function $f(x)$ (e.g., $f(x) + k$) affect the vertical movement, while changes inside the function (e.g., $f(x - h)$) affect the horizontal movement.