To find the coordinates of the image of quadrilateral QRST after being reflected over the line \( y = -1 \), we need to apply the reflection rules for each of the points \( Q(3, 1) \), \( R(4, 3) \), \( S(5, 3) \), and \( T(4, 0) \).
The reflection of a point \( (x, y) \) over the line \( y = k \) can be found using the formula: \[ \text{Reflected point} = (x, 2k - y) \]
In this case, \( k = -1 \).
Now we will find the reflected coordinates for each of the points:
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For point \( Q(3, 1) \): \[ Q' = (3, 2(-1) - 1) = (3, -2 - 1) = (3, -3) \]
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For point \( R(4, 3) \): \[ R' = (4, 2(-1) - 3) = (4, -2 - 3) = (4, -5) \]
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For point \( S(5, 3) \): \[ S' = (5, 2(-1) - 3) = (5, -2 - 3) = (5, -5) \]
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For point \( T(4, 0) \): \[ T' = (4, 2(-1) - 0) = (4, -2 - 0) = (4, -2) \]
Now we have the new coordinates of the image quadrilateral \( Q'R'S'T' \):
- \( Q'(3, -3) \)
- \( R'(4, -5) \)
- \( S'(5, -5) \)
- \( T'(4, -2) \)
Thus, the coordinates of the image \( QRST \) after reflection over the line \( y = -1 \) are \( Q'(3, -3) \), \( R'(4, -5) \), \( S'(5, -5) \), and \( T'(4, -2) \).