Answer :
To determine which set of transformations match the required transformations involving reflections, rotations, and dilations, we will analyze each transformation step-by-step. By examining the effect of each transformation, we can then identify which sequences of transformations are viable.
### Case 1: Reflection over the x-axis, then dilation by a scale factor of 3
1. Reflection over the x-axis: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
2. Dilation by a scale factor of 3: This transformation scales the coordinates by 3, changing [tex]\((x, -y)\)[/tex] to [tex]\((3x, -3y)\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (3x, -3y)\)[/tex] is achieved.
### Case 2: Reflection over the x-axis, then dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]
1. Reflection over the x-axis: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
2. Dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]: This transformation scales the coordinates by [tex]\(\frac{1}{3}\)[/tex], changing [tex]\((x, -y)\)[/tex] to [tex]\((\frac{x}{3}, -\frac{y}{3})\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (\frac{x}{3}, -\frac{y}{3})\)[/tex] is achieved.
### Case 3: 180-degree rotation about the origin, then dilation by a scale factor of 3
1. 180-degree rotation about the origin: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
2. Dilation by a scale factor of 3: This transformation scales the coordinates by 3, changing [tex]\((-x, -y)\)[/tex] to [tex]\((-3x, -3y)\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (-3x, -3y)\)[/tex] is achieved.
### Case 4: 180-degree rotation about the origin, then dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]
1. 180-degree rotation about the origin: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
2. Dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]: This transformation scales the coordinates by [tex]\(\frac{1}{3}\)[/tex], changing [tex]\((-x, -y)\)[/tex] to [tex]\((- \frac{x}{3}, - \frac{y}{3})\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (- \frac{x}{3}, - \frac{y}{3})\)[/tex] is achieved.
### Conclusion
All the given transformations (Cases 1, 2, 3, and 4) are preserved through the steps described, and they all match the required effect.
Therefore, the transformations that could be performed are:
1. A reflection over the x-axis, then a dilation by a scale factor of 3.
2. A reflection over the x-axis, then a dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex].
3. A 180-degree rotation about the origin, then a dilation by a scale factor of 3.
4. A 180-degree rotation about the origin, then a dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex].
These transformations adequately achieve the desired results, so the list of viable transformation sequences is [tex]\([1, 2, 3, 4]\)[/tex].
### Case 1: Reflection over the x-axis, then dilation by a scale factor of 3
1. Reflection over the x-axis: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
2. Dilation by a scale factor of 3: This transformation scales the coordinates by 3, changing [tex]\((x, -y)\)[/tex] to [tex]\((3x, -3y)\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (3x, -3y)\)[/tex] is achieved.
### Case 2: Reflection over the x-axis, then dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]
1. Reflection over the x-axis: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
2. Dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]: This transformation scales the coordinates by [tex]\(\frac{1}{3}\)[/tex], changing [tex]\((x, -y)\)[/tex] to [tex]\((\frac{x}{3}, -\frac{y}{3})\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (\frac{x}{3}, -\frac{y}{3})\)[/tex] is achieved.
### Case 3: 180-degree rotation about the origin, then dilation by a scale factor of 3
1. 180-degree rotation about the origin: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
2. Dilation by a scale factor of 3: This transformation scales the coordinates by 3, changing [tex]\((-x, -y)\)[/tex] to [tex]\((-3x, -3y)\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (-3x, -3y)\)[/tex] is achieved.
### Case 4: 180-degree rotation about the origin, then dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]
1. 180-degree rotation about the origin: This transformation changes a point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex].
2. Dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex]: This transformation scales the coordinates by [tex]\(\frac{1}{3}\)[/tex], changing [tex]\((-x, -y)\)[/tex] to [tex]\((- \frac{x}{3}, - \frac{y}{3})\)[/tex].
So the transformation [tex]\((x, y) \rightarrow (- \frac{x}{3}, - \frac{y}{3})\)[/tex] is achieved.
### Conclusion
All the given transformations (Cases 1, 2, 3, and 4) are preserved through the steps described, and they all match the required effect.
Therefore, the transformations that could be performed are:
1. A reflection over the x-axis, then a dilation by a scale factor of 3.
2. A reflection over the x-axis, then a dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex].
3. A 180-degree rotation about the origin, then a dilation by a scale factor of 3.
4. A 180-degree rotation about the origin, then a dilation by a scale factor of [tex]\(\frac{1}{3}\)[/tex].
These transformations adequately achieve the desired results, so the list of viable transformation sequences is [tex]\([1, 2, 3, 4]\)[/tex].
