Question

mild: check your understanding 5) translation: 4 left and 2 up 5) rotation: 180° about the origin 6) reflection: across the x - axis describe the transformation that occurred to get from shape one to shape two. medium: 2) 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: Understand translation rules

For a translation \(a\) units left and \(b\) units up, the transformation rule for a point \((x,y)\) is \((x - a,y + b)\). For a translation \(a\) units left and \(b\) units down, the rule is \((x - a,y - b)\).

Step2: Understand rotation rules

A \(180^{\circ}\) rotation about the origin for a point \((x,y)\) gives \((-x,-y)\). A \(90^{\circ}\) clock - wise rotation about the origin for a point \((x,y)\) gives \((y,-x)\).

Step3: Understand reflection rules

A reflection across the \(x\) - axis for a point \((x,y)\) gives \((x,-y)\). A reflection across the \(y\) - axis for a point \((x,y)\) gives \((-x,y)\).

Step4: Analyze each problem

• For "Translation: 4 left and 2 up", apply \((x,y)\to(x - 4,y + 2)\) to each vertex of the shape.
• For "Rotation: \(180^{\circ}\) about the origin", apply \((x,y)\to(-x,-y)\) to each vertex.
• For "Reflection: Across the \(x\) - axis", apply \((x,y)\to(x,-y)\) to each vertex.
• For "Translation: 2 left and 4 down", apply \((x,y)\to(x - 2,y - 4)\) to each vertex.
• For "Rotation: \(90^{\circ}\) clockwise about the origin", apply \((x,y)\to(y,-x)\) to each vertex.
• For "Reflection: Across the \(y\) - axis", apply \((x,y)\to(-x,y)\) to each vertex.

For the "Describe the transformation" parts:

• For the first "Describe the transformation" (top - right of the second section):
• Compare the coordinates of the vertices of shape 1 and shape 2. If we observe the change in \(x\) and \(y\) values of corresponding vertices. Suppose a vertex of shape 1 is \((x_1,y_1)\) and of shape 2 is \((x_2,y_2)\). Analyze the pattern. If \(x_2=x_1 - h\) and \(y_2=y_1 - k\), it is a translation \(h\) units left and \(k\) units down. If \(x_2=-x_1\) and \(y_2 = y_1\), it is a reflection across the \(y\) - axis. If \(x_2=x_1\) and \(y_2=-y_1\), it is a reflection across the \(x\) - axis. If \(x_2=-x_1\) and \(y_2=-y_1\), it is a \(180^{\circ}\) rotation about the origin. If \(x_2 = y_1\) and \(y_2=-x_1\), it is a \(90^{\circ}\) clock - wise rotation about the origin.
• For the second "Describe the transformation" (bottom - right of the second section):
• Similarly, compare the coordinates of the vertices of shape 1 and shape 2 of the trapezoid - like shapes. Identify if it is a translation (by looking at the change in \(x\) and \(y\) values of corresponding vertices), a rotation (by checking the orientation change and using the rotation rules for coordinates), or a reflection (by seeing if the shape is flipped over an axis).

Since no specific vertices or final - state shapes are given to draw, the above steps show how to perform and describe the geometric transformations.

Answer:

The steps above show how to perform and describe the given geometric transformations of translation, rotation, and reflection. For the "Describe the transformation" parts, compare vertex coordinates of the two shapes to identify the transformation type.