To graph the transformation for the function $g(x) = \frac{1}{2}f(x + 3) - 1$, we need to break down the effects of each transformation applied to the base function $f(x)$, which is given by the graph in your image. Here’s a step-by-step explanation of each transformation:

1. Horizontal Shift:

   • The term $f(x + 3)$ shifts the graph of $f(x)$ to the left by 3 units. This is because adding 3 inside the function argument shifts the graph horizontally in the opposite direction.

2. Vertical Compression:

   • The factor $\frac{1}{2}$ multiplies the function, which results in a vertical compression by a factor of $\frac{1}{2}$. This makes the graph flatter by halving the distance of all points from the x-axis.

3. Vertical Shift:

   • The term $-1$ outside the function shifts the entire graph down by 1 unit.

Steps to Graph $g(x)$:

• Start with the graph of $f(x)$ (the one provided in the image).
• Shift the graph 3 units to the left to account for the horizontal shift.
• Compress the graph vertically by a factor of $\frac{1}{2}$. This means all y-values of the points on the graph of $f(x)$ will be halved.
• Shift the graph 1 unit down to apply the vertical shift.

Once these steps are applied, the new graph will represent the transformed function $g(x)$.

Would you like me to create an illustration of the graph after these transformations?

Follow-up Questions:

1. How do horizontal and vertical shifts differ in terms of transformation rules?
2. What is the effect of multiplying a function by a fraction less than 1?
3. Can you describe the changes made to the maximum and minimum points after compression?
4. How does shifting left differ from shifting right in function transformations?
5. How does the vertical shift affect the range of the function?

Tip: Always apply horizontal transformations (inside the function) before vertical ones (outside the function).