Question

jean transformed a point by using the rule \\((x, y) \
ightarrow (x - 6, y + 8)\\). the image point is \\((-4, 1)\\). which point is the pre-image?

\\((-10, 9)\\)
\\((2, -7)\\)
\\((-2, 7)\\)
\\((10, -9)\\)

Explanation:

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"Translation Rule",
"Coordinate Transformations"
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"current_concepts": [
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"Coordinate Transformations"
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</pre_analysis>

<reasoning>

Set up the transformation equations

Using the Translation Rule and Coordinate Transformations knowledge points
\[

$$\begin{aligned} &x - 6 = -4 \\ &y + 8 = 1 \end{aligned}$$

\]

Solve for the pre-image coordinates

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &x = -4 + 6 = 2 \\ &y = 1 - 8 = -7 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((-10, 9)\)</mcq-option>
<mcq-correct>(B) \((2, -7)\)</mcq-correct>
<mcq-option>(C) \((-2, 7)\)</mcq-option>
<mcq-option>(D) \((10, -9)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[2, -7]],
"properties": {
"name": "Pre-image (2, -7)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-4, 1]],
"properties": {
"name": "Image (-4, 1)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "arrow",
"params": [[2, -7], [-4, 1]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
}
]
}
</plot>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Coordinate Transformations"
]
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</post_analysis>

Answer:

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"Coordinate Transformations"
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</pre_analysis>

<reasoning>

Set up the transformation equations

Using the Translation Rule and Coordinate Transformations knowledge points
\[

$$\begin{aligned} &x - 6 = -4 \\ &y + 8 = 1 \end{aligned}$$

\]

Solve for the pre-image coordinates

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &x = -4 + 6 = 2 \\ &y = 1 - 8 = -7 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((-10, 9)\)</mcq-option>
<mcq-correct>(B) \((2, -7)\)</mcq-correct>
<mcq-option>(C) \((-2, 7)\)</mcq-option>
<mcq-option>(D) \((10, -9)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[2, -7]],
"properties": {
"name": "Pre-image (2, -7)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-4, 1]],
"properties": {
"name": "Image (-4, 1)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "arrow",
"params": [[2, -7], [-4, 1]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
}
]
}
</plot>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
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"Coordinate Transformations"
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