The task requires graphing the function $y = -2f(x-1) - 2$ using the graph of $y = f(x)$. The points provided are based on transformations applied to the original function $f(x)$. Let’s break down the transformations:

• The $(x-1)$ represents a horizontal shift to the right by 1 unit.
• The multiplication by $-2$ reflects the graph vertically (because of the negative sign) and stretches it by a factor of 2.
• Finally, the $-2$ represents a vertical shift down by 2 units.

Given Points and Transformation:

You are provided with the following points on $f(x)$:

1. $(-4, -4)$
2. $(-1, -3)$
3. $(0, 0)$
4. $(1, -4)$
5. $(3, 3)$

Let's apply the transformation $y = -2f(x-1) - 2$ to these points:

1. Point (-4, -4): $\text{Shift } x = -4 + 1 = -3, \quad y = -2(-4) - 2 = 8 - 2 = 6$ Transformed point: $(-3, 6)$

2. Point (-1, -3): $\text{Shift } x = -1 + 1 = 0, \quad y = -2(-3) - 2 = 6 - 2 = 4$ Transformed point: $(0, 4)$

3. Point (0, 0): $\text{Shift } x = 0 + 1 = 1, \quad y = -2(0) - 2 = 0 - 2 = -2$ Transformed point: $(1, -2)$

4. Point (1, -4): $\text{Shift } x = 1 + 1 = 2, \quad y = -2(-4) - 2 = 8 - 2 = 6$ Transformed point: $(2, 6)$

5. Point (3, 3): $\text{Shift } x = 3 + 1 = 4, \quad y = -2(3) - 2 = -6 - 2 = -8$ Transformed point: $(4, -8)$

Choose the Correct Graph:

Now, compare the transformed points with the graphs (A and B). Based on the transformation:

• The point $(1, -2)$ suggests the curve will go below the $x$-axis.
• Points like $(0, 4)$ and $(2, 6)$ suggest the graph will have a stretch above the $x$-axis.

From these clues, Graph B is the correct representation, as it matches the transformations we’ve computed.

Do you want further details on the transformation process or the points?

Relative questions:

1. How do horizontal and vertical shifts affect the graph of a function?
2. What is the effect of a vertical stretch by a factor of 2 on a graph?
3. How does a reflection across the x-axis change the graph?
4. How do you determine the correct graph after applying transformations?
5. What would happen if you reflected the graph across the y-axis instead?

Tip: Always apply transformations step-by-step, especially when multiple transformations are involved.