To determine which transformations will map a figure onto itself, we need to consider the geometric properties of the transformations in relation to the shape in question. However, since the specific figure is not provided, I can explain the typical effects of each transformation:
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Translation up 4 units: This will not map a figure onto itself unless the figure is specifically positioned such that it can overlap with its original position. For most figures, this will not hold.
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Reflection across the y-axis: This transformation can map some figures onto themselves, particularly symmetrical figures that have a line of symmetry along the y-axis (like a circle or a vertically symmetrical shape).
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Dilation by a factor of 3 about the origin: Dilation changes the size of the figure and will not map any figure onto itself (it will either expand or contract the figure).
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Rotation 90 degrees clockwise about the origin: This transformation will map a figure onto itself for figures that are symmetric or have specific rotational symmetry (like squares or circles) but generally will not for most shapes.
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Rotation 180 degrees counterclockwise about the origin: This transformation can often map figures onto themselves if they have point symmetry, such as a rectangle or a circle.
In summary:
- Reflection across the y-axis: May map onto itself depending on symmetry.
- Rotation 180 degrees counterclockwise about the origin: May also map onto itself depending on symmetry.
The transformations that generally would map the figure onto itself would be only the reflection across the y-axis and the rotation 180 degrees counterclockwise about the origin, assuming the figure is symmetric in those ways. The translations and dilation would not map the figure back onto itself.