Question

1. the graph of a linear function is shown on the grid. write the equation that is best represented by the graph.

slope-intercept form point-slope form standard form

Explanation:

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"Point-Slope Form",
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<reasoning>

Identify key points on the graph

\[

$$\begin{aligned} &\text{y-intercept: } (0, 2)\\ &\text{x-intercept: } (-3, 0)\\ &\text{Another integer point: } (3, 4) \end{aligned}$$

\]

Calculate the slope

\[
m = \frac{2 - 0}{0 - (-3)} = \frac{2}{3}
\]

Write the linear equations in different forms

\[

$$\begin{aligned} &\text{Slope-intercept form: } y = \frac{2}{3}x + 2\\ &\text{Point-slope form: } y - 2 = \frac{2}{3}(x - 0) \quad \text{or} \quad y - 0 = \frac{2}{3}(x + 3)\\ &\text{Standard form: } 2x - 3y = -6 \end{aligned}$$

\]
</reasoning>

<answer>

• Slope-intercept form: \(y = \frac{2}{3}x + 2\)
• Point-slope form: \(y - 2 = \frac{2}{3}x\) (or \(y = \frac{2}{3}(x + 3)\))
• Standard form: \(2x - 3y = -6\)

</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 2]],
"properties": {
"name": "(0, 2)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-3, 0]],
"properties": {
"name": "(-3, 0)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "(2/3)*x + 2", "latex": "\frac{2}{3}x + 2"}, -10, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Standard Form of a Linear Equation"
]
}
</post_analysis>

Answer:

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"step_cot",
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],
"needs_drawing": true,
"concepts_used": [
"Slope-Intercept Form",
"Point-Slope Form",
"y-intercept"
],
"new_concepts": [
"Standard Form of a Linear Equation"
],
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"Slope-Intercept Form",
"Point-Slope Form",
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"y-intercept"
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}
</pre_analysis>

<reasoning>

Identify key points on the graph

\[

$$\begin{aligned} &\text{y-intercept: } (0, 2)\\ &\text{x-intercept: } (-3, 0)\\ &\text{Another integer point: } (3, 4) \end{aligned}$$

\]

Calculate the slope

\[
m = \frac{2 - 0}{0 - (-3)} = \frac{2}{3}
\]

Write the linear equations in different forms

\[

$$\begin{aligned} &\text{Slope-intercept form: } y = \frac{2}{3}x + 2\\ &\text{Point-slope form: } y - 2 = \frac{2}{3}(x - 0) \quad \text{or} \quad y - 0 = \frac{2}{3}(x + 3)\\ &\text{Standard form: } 2x - 3y = -6 \end{aligned}$$

\]
</reasoning>

<answer>

• Slope-intercept form: \(y = \frac{2}{3}x + 2\)
• Point-slope form: \(y - 2 = \frac{2}{3}x\) (or \(y = \frac{2}{3}(x + 3)\))
• Standard form: \(2x - 3y = -6\)

</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 2]],
"properties": {
"name": "(0, 2)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-3, 0]],
"properties": {
"name": "(-3, 0)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "(2/3)*x + 2", "latex": "\frac{2}{3}x + 2"}, -10, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Standard Form of a Linear Equation"
]
}
</post_analysis>