To determine which transformations will map the figure onto itself, we need to consider each transformation option one by one:
A) Translation up 4 units: This will move every point of the figure up by 4 units. Therefore, the figure will not map onto itself and this option is not correct.
B) Reflection across the y-axis: Reflecting the figure across the y-axis means that all points will swap their x-coordinates to their negatives. If the figure is symmetric concerning the y-axis, it will map onto itself. If it is not symmetric about the y-axis, this will not hold. We cannot conclusively say that this transformation will map the figure onto itself without seeing the specific figure.
C) Dilation by a factor of 3 about the origin: Dilation increases the size of the figure. Thus, the figure will not map onto itself as its dimensions will change. This option is not correct.
D) Rotation 90 degrees clockwise about the origin: Rotating the figure by 90 degrees clockwise will generally change its orientation unless it has certain symmetrical properties that allow it to map onto itself. Thus, this option is likely not correct, but again, it depends on the specific figure.
E) Rotation 180 degrees counterclockwise about the origin: Rotating the figure 180 degrees will often map a figure onto itself, especially if it has rotational symmetry. This is a common transformation for many geometric figures, so it's likely that this option is correct, but relies on the figure's specific characteristics.
In conclusion:
- B and E are potential candidates that could map the figure onto itself, depending on the figure's symmetry properties.
- A, C, and D are confirmed as not correct.
If you need to be more specific about the answer based on the figure you are referring to, please provide more details about the figure's shape or symmetry.