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Question

A coordinate plane's axes range from negative 6 to 6, both by 1-unit increments. Points upper L, upper O, and upper W are plotted and connected via lines that form a triangle.

Which quadrant will ΔLOW be in when it is reflected across the y-axis and then reflected across the x-axis?

(1 point)
Responses

Quadrant I
Quadrant I

Quadrant II
Quadrant II

Quadrant III
Quadrant III

Quadrant IV
Quadrant IV
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Answer 1

To determine the quadrant of triangle ΔLOW after the specified reflections, we need to understand how reflections across the axes affect the coordinates of points.

1. Reflection across the y-axis: When a point (x, y) is reflected across the y-axis, its new coordinates become (-x, y). This means the x-coordinate changes sign while the y-coordinate remains the same.

2. Reflection across the x-axis: When a point (x, y) is reflected across the x-axis, its new coordinates become (x, -y). This means the y-coordinate changes sign while the x-coordinate remains the same.

Now, follow these steps:

1. Take any point (x, y) from triangle ΔLOW.
2. Reflect across the y-axis: This transforms the point to (-x, y).
3. Now reflect this new point across the x-axis: This transforms it to (-x, -y).

These transformations result in a new point (-x, -y).

Next, we need to determine the quadrant of the resulting point (-x, -y). The quadrants are defined as follows:

• Quadrant I: (x, y) where both x and y are positive.
• Quadrant II: (x, y) where x is negative and y is positive.
• Quadrant III: (x, y) where both x and y are negative.
• Quadrant IV: (x, y) where x is positive and y is negative.

Since both x and y are changed to their negatives in the final point (-x, -y), if the original point (x, y) was in Quadrant I, it would move to Quadrant III after the reflections because both coordinates become negative.

Therefore, after reflecting triangle ΔLOW across the y-axis and then across the x-axis, it will be in:

Quadrant III.