Question

1. translation: 4 left and 2 up
2. rotation: 180° about the origin
3. reflection: across the x - axis

describe the transformation that occurred to get from shape one to shape two.

1. translation: 2 left and 4 down
2. rotation: 90° clockwise about the origin
3. reflection: across the y - axis

describe the transformation that occurred to get from shape one to shape two.

Explanation:

Step1: Recall transformation rules

Translation: For a point $(x,y)$, a translation of $a$ units left and $b$ units up gives $(x - a,y + b)$. Rotation of $180^{\circ}$ about the origin for a point $(x,y)$ gives $(-x,-y)$. Reflection across the $x$ - axis for a point $(x,y)$ gives $(x,-y)$ and reflection across the $y$ - axis gives $(-x,y)$. Rotation of $90^{\circ}$ clockwise about the origin for a point $(x,y)$ gives $(y,-x)$.

Step2: Analyze each transformation

For "Translation: 4 left and 2 up", we subtract 4 from the $x$ - coordinate and add 2 to the $y$ - coordinate of each point of the shape. For "Rotation: 180^{\circ} about the origin", we change the sign of both coordinates of each point. For "Reflection: Across the $x$ - axis", we change the sign of the $y$ - coordinate of each point. For "Translation: 2 left and 4 down", we subtract 2 from the $x$ - coordinate and subtract 4 from the $y$ - coordinate of each point. For "Rotation: 90^{\circ} clockwise about the origin", we swap the $x$ and $y$ coordinates and change the sign of the new $x$ - coordinate. For "Reflection: Across the $y$ - axis", we change the sign of the $x$ - coordinate of each point.

Step3: Describe the transformation for the last two un - labeled cases

For the first un - labeled case (bottom - right), comparing shape 1 and shape 2, we can see that each point $(x,y)$ of shape 1 has been transformed to $(x,y - 3)$. So the transformation is a translation 3 units down. For the second un - labeled case (middle - right), comparing shape 1 and shape 2, each point $(x,y)$ of shape 1 has been transformed to $(-x,y)$. So the transformation is a reflection across the $y$ - axis.

Answer:

For the un - labeled transformation on the bottom - right: Translation 3 units down. For the un - labeled transformation on the middle - right: Reflection across the $y$ - axis.