Question

which is the image of triangle abc after this sequence of transformations:

1. reflection across the y-axis
2. dilation by a factor of 0.5 centered at the origin

select the correct figure on the graph.

Explanation:

Identify the coordinates of the original triangle \(ABC\)

We first find the coordinates of the vertices of triangle \(ABC\) from the graph:

• Vertex \(A\) is located at \((2, -2)\).
• Vertex \(B\) is located at \((6, -4)\).
• Vertex \(C\) is located at \((4, -4)\).

Apply the reflection across the y-axis

Using the rule for reflection across the y-axis, \((x, y)
ightarrow (-x, y)\), we transform each vertex:

• \(A(2, -2)

ightarrow A'(-2, -2)\)

• \(B(6, -4)

ightarrow B'(-6, -4)\)

• \(C(4, -4)

ightarrow C'(-4, -4)\)

Apply the dilation by a factor of 0.5 centered at the origin

Using the Dilation in Coordinate Plane rule, \((x, y)
ightarrow (0.5x, 0.5y)\), we transform the reflected vertices:

• \(A'(-2, -2)

ightarrow A''(-1, -1)\)

• \(B'(-6, -4)

ightarrow B''(-3, -2)\)

• \(C'(-4, -4)

ightarrow C''(-2, -2)\)

Identify the matching figure on the graph

We look for the triangle with vertices at \((-1, -1)\), \((-3, -2)\), and \((-2, -2)\):

• The blue/orange triangle in the third quadrant has vertices at \((-1, -1)\), \((-5, -4)\), and \((-3, -4)\) (which represents only the reflection and not the correct dilation).
• The other blue/orange triangle in the third quadrant has vertices at \((-1, -1)\), \((-3, -2)\), and \((-2, -2)\).
• Therefore, the smaller triangle in the third quadrant with vertices at \((-1, -1)\), \((-3, -2)\), and \((-2, -2)\) is the correct image.

Answer:

The correct figure on the graph is the small orange and blue triangle located in the third quadrant with vertices at \((-1, -1)\), \((-3, -2)\), and \((-2, -2)\).