The given function transformation is $g(x) = -3f(x - 1) + 1$, where $f(x) = x$ is the parent function.

To graph the transformation:

1. Horizontal Shift: The term $(x - 1)$ indicates a shift 1 unit to the right.
2. Vertical Stretch/Reflection: The coefficient $-3$ means the graph is vertically stretched by a factor of 3 and reflected over the x-axis.
3. Vertical Shift: The $+1$ moves the entire graph 1 unit up.

Step-by-step breakdown:

• Start by shifting the line $f(x) = x$ 1 unit to the right.
• Then apply the vertical stretch and reflection by multiplying the slope by -3 (i.e., slope becomes -3).
• Finally, move the graph up by 1 unit.

Would you like a detailed step-by-step plotting of the points, or have any specific questions?

5 Related Questions:

1. What effect does a negative coefficient have on a function's graph?
2. How does a horizontal shift affect the domain of a function?
3. What is the difference between a vertical stretch and compression?
4. Can you explain how transformations apply to non-linear functions like quadratics?
5. How do combined transformations (like shift and reflection) affect the overall graph?

Tip:

Remember, transformations like stretches and shifts can be applied in sequence, but order matters—typically, handle horizontal transformations before vertical ones!