Question

which graph matches this direct variation equation?
\\y = -2x\\

Explanation:

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<reasoning>

Identify key features of the equation

\[

$$\begin{aligned} &y = -2x \\ &\text{Slope } (m) = -2 \\ &\text{y-intercept } (b) = 0 \end{aligned}$$

\]

Determine points on the graph

\[

$$\begin{aligned} &x = 0 \implies y = 0 \quad (0,0) \\ &x = 1 \implies y = -2 \quad (1,-2) \\ &x = -1 \implies y = 2 \quad (-1,2) \end{aligned}$$

\]

Match points with the given options

\[

$$\begin{aligned} &\text{Graph 1 (Left): passes through } (0,0), (-1,2), (1,-2) \\ &\text{Graph 2 (Middle): passes through } (0,0), (1,2), (-1,-2) \\ &\text{Graph 3 (Right): passes through } (0,0), (-1,3), (1,-3) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) The first graph (left) with points at (-1, 2), (0, 0), and (1, -2)</mcq-correct>
<mcq-option>(B) The second graph (middle) with points at (1, 2), (0, 0), and (-1, -2)</mcq-option>
<mcq-option>(C) The third graph (right) with points at (-1, 3), (0, 0), and (1, -3)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "-2*x",
"latex": "y = -2x"
},
-5,
5
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3,
"name": "y = -2x",
"withLabel": true
}
},
{
"type": "point",
"params": [
[0, 0]
],
"properties": {
"name": "(0,0)",
"size": 4,
"color": "#5583F2",
"strokeColor": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [
[-1, 2]
],
"properties": {
"name": "(-1,2)",
"size": 4,
"color": "#5583F2",
"strokeColor": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [
[1, -2]
],
"properties": {
"name": "(1,-2)",
"size": 4,
"color": "#5583F2",
"strokeColor": "#5583F2",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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"Mathematics",
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</post_analysis>

Answer:

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],
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"concepts_used": [
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],
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</pre_analysis>

<reasoning>

Identify key features of the equation

\[

$$\begin{aligned} &y = -2x \\ &\text{Slope } (m) = -2 \\ &\text{y-intercept } (b) = 0 \end{aligned}$$

\]

Determine points on the graph

\[

$$\begin{aligned} &x = 0 \implies y = 0 \quad (0,0) \\ &x = 1 \implies y = -2 \quad (1,-2) \\ &x = -1 \implies y = 2 \quad (-1,2) \end{aligned}$$

\]

Match points with the given options

\[

$$\begin{aligned} &\text{Graph 1 (Left): passes through } (0,0), (-1,2), (1,-2) \\ &\text{Graph 2 (Middle): passes through } (0,0), (1,2), (-1,-2) \\ &\text{Graph 3 (Right): passes through } (0,0), (-1,3), (1,-3) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) The first graph (left) with points at (-1, 2), (0, 0), and (1, -2)</mcq-correct>
<mcq-option>(B) The second graph (middle) with points at (1, 2), (0, 0), and (-1, -2)</mcq-option>
<mcq-option>(C) The third graph (right) with points at (-1, 3), (0, 0), and (1, -3)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "-2*x",
"latex": "y = -2x"
},
-5,
5
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3,
"name": "y = -2x",
"withLabel": true
}
},
{
"type": "point",
"params": [
[0, 0]
],
"properties": {
"name": "(0,0)",
"size": 4,
"color": "#5583F2",
"strokeColor": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [
[-1, 2]
],
"properties": {
"name": "(-1,2)",
"size": 4,
"color": "#5583F2",
"strokeColor": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [
[1, -2]
],
"properties": {
"name": "(1,-2)",
"size": 4,
"color": "#5583F2",
"strokeColor": "#5583F2",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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