Question

which image is a reflection of the black polygon in quadrant i?

the black polygon in quadrant i
the red polygon in quadrant ii
the blue polygon in quadrant iii

Explanation:

Identify the original figure and its vertices

The original figure is the black polygon located in Quadrant I. Let's identify the coordinates of its key vertices from the grid:

• Bottom-left vertex: \((2, 1)\)
• Bottom-right vertex: \((8, 1)\)
• Top-right outer corner: \((8, 3)\)
• Top-right inner corner: \((7, 3)\)
• Top-right inner dip: \((7, 2)\)
• Top-left peak: \((3, 3)\)

Analyze the transformation of the red polygon in Quadrant II

Using the Coordinate Transformations knowledge point, we check the coordinates of the red polygon in Quadrant II:

• Bottom-right vertex: \((-2, 1)\)
• Bottom-left vertex: \((-8, 1)\)
• Top-left outer corner: \((-8, 3)\)
• Top-left inner corner: \((-7, 3)\)
• Top-left inner dip: \((-7, 2)\)
• Top-right peak: \((-3, 3)\)

Comparing these to the original black polygon vertices \((x, y)\), each corresponding vertex is at \((-x, y)\). This represents a reflection across the y-axis.

Analyze the transformation of the blue polygon in Quadrant III

We check the coordinates of the blue polygon in Quadrant III:

• Top-right vertex: \((-2, -1)\)
• Top-left vertex: \((-8, -1)\)
• Bottom-left outer corner: \((-8, -3)\)
• Bottom-left inner corner: \((-7, -3)\)
• Bottom-left inner dip: \((-7, -2)\)
• Bottom-right peak: \((-3, -3)\)

Comparing these to the original black polygon vertices \((x, y)\), each corresponding vertex is at \((-x, -y)\). This represents a rotation of \(180^\circ\) about the origin (or a double reflection across both axes).

Analyze the transformation of the green polygon in Quadrant IV

We check the coordinates of the green polygon in Quadrant IV:

• Top-left vertex: \((2, -1)\)
• Top-right vertex: \((8, -1)\)
• Bottom-right outer corner: \((8, -3)\)
• Bottom-right inner corner: \((7, -3)\)
• Bottom-right inner dip: \((7, -2)\)
• Bottom-left peak: \((3, -3)\)

Comparing these to the original black polygon vertices \((x, y)\), each corresponding vertex is at \((x, -y)\). This represents a reflection across the x-axis.

Determine the correct option

The question asks "Which image is a reflection of the black polygon in quadrant I?".

• The red polygon in quadrant II is a reflection across the y-axis.
• The green polygon in quadrant IV is a reflection across the x-axis.

Looking at the visible multiple-choice options:

1. "the black polygon in quadrant I" (Original image)
2. "the red polygon in quadrant II" (Valid reflection across the y-axis)
3. "the blue polygon in quadrant III" (Rotation, not a single reflection)

The fourth option (partially cut off but representing the green polygon in quadrant IV) is not fully visible, but "the red polygon in quadrant II" is a clearly listed, correct option representing a reflection.

Answer:

• the black polygon in quadrant I
• the red polygon in quadrant II (Correct answer)
• the blue polygon in quadrant III