To analyze the transformations and where the final image \( \triangle Q''R''S'' \) will be located, we will first apply the transformations step by step.
Step 1: Apply the first transformation to triangle QRS:
- The coordinates of triangle \( QRS \) are:
- \( Q(-8, 4) \)
- \( R(-5, 8) \)
- \( S(-3, 3) \)
Applying the transformation \( (x,y) \rightarrow (x+4, y-6) \):
- New coordinates for \( Q' \): \[ Q' = (-8 + 4, 4 - 6) = (-4, -2) \]
- New coordinates for \( R' \): \[ R' = (-5 + 4, 8 - 6) = (-1, 2) \]
- New coordinates for \( S' \): \[ S' = (-3 + 4, 3 - 6) = (1, -3) \]
Now, the new coordinates for \( \triangle Q'R'S' \) are \( Q'(-4, -2), R'(-1, 2), S'(1, -3) \).
Step 2: Apply the second transformation to triangle \( Q'R'S' \): Applying the transformation \( (x,y) \rightarrow (x+5, y-3) \):
- New coordinates for \( Q'' \): \[ Q'' = (-4 + 5, -2 - 3) = (1, -5) \]
- New coordinates for \( R'' \): \[ R'' = (-1 + 5, 2 - 3) = (4, -1) \]
- New coordinates for \( S'' \): \[ S'' = (1 + 5, -3 - 3) = (6, -6) \]
The new coordinates for \( \triangle Q''R''S'' \) are \( Q''(1, -5), R''(4, -1), S''(6, -6) \).
a. Determining the quadrant of \( \triangle Q''R''S'' \):
The coordinates \( (1, -5) \), \( (4, -1) \), and \( (6, -6) \):
- All these points have a positive x-coordinate and a negative y-coordinate.
- Therefore, all points lie in the fourth quadrant.
b. Finding the rule for the composition transformation:
To find the overall transformation from \( (x, y) \) to \( (x', y') \), we need to combine the two transformations step-by-step:
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First transformation: \[ \begin{align*} x' &= x + 4 \ y' &= y - 6 \end{align*} \]
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Second transformation using the result of the first: \[ \begin{align*} x'' &= (x + 4) + 5 = x + 9 \ y'' &= (y - 6) - 3 = y - 9 \end{align*} \]
The overall transformation from \( (x, y) \) to \( (x'', y'') \) is: \[ (x, y) \rightarrow (x + 9, y - 9) \]
Thus, the rule for the composition transformation is \( (x + 9, y - 9) \).
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