Solved on Dec 12, 2023

Find the image coordinates of a point $Q(-5,1)$ under the transformation $(x, y) \rightarrow(-y,-x)$ .

STEP 1

Assumptions
1. The preimage coordinate is $Q(-5,1)$ .
2. The transformation is described by the coordinate notation $(x, y) \rightarrow(-y,-x)$ .
3. We need to apply the transformation to the preimage to find the image coordinate $Q'$ .

STEP 2

Identify the components of the preimage coordinate.
The preimage coordinate is $Q(-5,1)$ , where $x = -5$ and $y = 1$ .

STEP 3

Apply the transformation to the preimage coordinate.
The transformation $(x, y) \rightarrow(-y,-x)$ tells us to replace $x$ with $-y$ and $y$ with $-x$ .

STEP 4

Substitute the values of $x$ and $y$ from the preimage into the transformation.
For $Q(-5,1)$ , we have $x = -5$ and $y = 1$ . Applying the transformation:
$(-y,-x) = (-(1),-(-5))$

STEP 5

Calculate the new coordinates after applying the transformation.
$(-y,-x) = (-1,5)$

STEP 6

Write down the coordinates of the image $Q'$ .
The image coordinate after the transformation is $Q'(-1,5)$ .
Therefore, the correct answer is $Q^{\prime}(-1,5)$ .

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Find the image coordinates of a point $Q(-5,1)$ under the transformation $(x, y) \rightarrow(-y,-x)$ .

STEP 1

Assumptions
1. The preimage coordinate is $Q(-5,1)$ .
2. The transformation is described by the coordinate notation $(x, y) \rightarrow(-y,-x)$ .
3. We need to apply the transformation to the preimage to find the image coordinate $Q'$ .

STEP 2

Identify the components of the preimage coordinate.
The preimage coordinate is $Q(-5,1)$ , where $x = -5$ and $y = 1$ .

STEP 3

Apply the transformation to the preimage coordinate.
The transformation $(x, y) \rightarrow(-y,-x)$ tells us to replace $x$ with $-y$ and $y$ with $-x$ .

STEP 4

Substitute the values of $x$ and $y$ from the preimage into the transformation.
For $Q(-5,1)$ , we have $x = -5$ and $y = 1$ . Applying the transformation:
$(-y,-x) = (-(1),-(-5))$

STEP 5

Calculate the new coordinates after applying the transformation.
$(-y,-x) = (-1,5)$

STEP 6

Write down the coordinates of the image $Q'$ .
The image coordinate after the transformation is $Q'(-1,5)$ .
Therefore, the correct answer is $Q^{\prime}(-1,5)$ .

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Find the image coordinates of a point Q(−5,1)Q(-5,1)Q(−5,1) under the transformation (x,y)→(−y,−x)(x, y) \rightarrow(-y,-x)(x,y)→(−y,−x).

STEP 1

Assumptions1. The preimage coordinate is Q(−5,1)Q(-5,1)Q(−5,1).2. The transformation is described by the coordinate notation (x,y)→(−y,−x)(x, y) \rightarrow(-y,-x)(x,y)→(−y,−x).3. We need to apply the transformation to the preimage to find the image coordinate Q′Q'Q′.

STEP 2

Identify the components of the preimage coordinate.The preimage coordinate is Q(−5,1)Q(-5,1)Q(−5,1), where x=−5x = -5x=−5 and y=1y = 1y=1.

STEP 3

Apply the transformation to the preimage coordinate.The transformation (x,y)→(−y,−x)(x, y) \rightarrow(-y,-x)(x,y)→(−y,−x) tells us to replace xxx with −y-y−y and yyy with −x-x−x.

STEP 4

Substitute the values of xxx and yyy from the preimage into the transformation.For Q(−5,1)Q(-5,1)Q(−5,1), we have x=−5x = -5x=−5 and y=1y = 1y=1. Applying the transformation:(−y,−x)=(−(1),−(−5))(-y,-x) = (-(1),-(-5))(−y,−x)=(−(1),−(−5))

STEP 5

Calculate the new coordinates after applying the transformation.(−y,−x)=(−1,5)(-y,-x) = (-1,5)(−y,−x)=(−1,5)

STEP 6

Write down the coordinates of the image Q′Q'Q′.The image coordinate after the transformation is Q′(−1,5)Q'(-1,5)Q′(−1,5).Therefore, the correct answer is Q′(−1,5)Q^{\prime}(-1,5)Q′(−1,5).

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Find the image coordinates of a point $Q(-5,1)$ under the transformation $(x, y) \rightarrow(-y,-x)$ .

Assumptions
1. The preimage coordinate is $Q(-5,1)$ .
2. The transformation is described by the coordinate notation $(x, y) \rightarrow(-y,-x)$ .
3. We need to apply the transformation to the preimage to find the image coordinate $Q'$ .

Identify the components of the preimage coordinate.
The preimage coordinate is $Q(-5,1)$ , where $x = -5$ and $y = 1$ .

Apply the transformation to the preimage coordinate.
The transformation $(x, y) \rightarrow(-y,-x)$ tells us to replace $x$ with $-y$ and $y$ with $-x$ .

Substitute the values of $x$ and $y$ from the preimage into the transformation.
For $Q(-5,1)$ , we have $x = -5$ and $y = 1$ . Applying the transformation:
$(-y,-x) = (-(1),-(-5))$

Calculate the new coordinates after applying the transformation.
$(-y,-x) = (-1,5)$

Write down the coordinates of the image $Q'$ .
The image coordinate after the transformation is $Q'(-1,5)$ .
Therefore, the correct answer is $Q^{\prime}(-1,5)$ .