Question

1. a quadratic function has a vertex at \\((9, -3)\\) and passes through \\((4, 97)\\).

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": true,
"concepts_used": [
"Quadratic Vertex Form",
"Determining Quadratic Equations"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Vertex Form",
"Determining Quadratic Equations"
]
}
</pre_analysis>

<reasoning>

Write the vertex form equation

Using the Quadratic Vertex Form knowledge point
\[
y = a(x - h)^2 + k
\]
\[
(h, k) = (9, -3)
\]
\[
y = a(x - 9)^2 - 3
\]

Substitute the given point to solve for a

Using the Determining Quadratic Equations knowledge point
\[
(x, y) = (4, 97)
\]
\[
97 = a(4 - 9)^2 - 3
\]
\[
97 = a(-5)^2 - 3
\]
\[
97 = 25a - 3
\]
\[
100 = 25a
\]
\[
a = 4
\]

Write the final quadratic equation

Using the Determining Quadratic Equations knowledge point
\[
y = 4(x - 9)^2 - 3
\]
</reasoning>

<answer>
\(y = 4(x - 9)^2 - 3\)
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[9, -3]],
"properties": {
"name": "Vertex (9, -3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 97]],
"properties": {
"name": "(4, 97)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [
{
"js": "4 * Math.pow(x - 9, 2) - 3",
"latex": "y = 4(x - 9)^2 - 3"
},
3,
15
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Vertex Form"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": true,
"concepts_used": [
"Quadratic Vertex Form",
"Determining Quadratic Equations"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Vertex Form",
"Determining Quadratic Equations"
]
}
</pre_analysis>

<reasoning>

Write the vertex form equation

Using the Quadratic Vertex Form knowledge point
\[
y = a(x - h)^2 + k
\]
\[
(h, k) = (9, -3)
\]
\[
y = a(x - 9)^2 - 3
\]

Substitute the given point to solve for a

Using the Determining Quadratic Equations knowledge point
\[
(x, y) = (4, 97)
\]
\[
97 = a(4 - 9)^2 - 3
\]
\[
97 = a(-5)^2 - 3
\]
\[
97 = 25a - 3
\]
\[
100 = 25a
\]
\[
a = 4
\]

Write the final quadratic equation

Using the Determining Quadratic Equations knowledge point
\[
y = 4(x - 9)^2 - 3
\]
</reasoning>

<answer>
\(y = 4(x - 9)^2 - 3\)
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[9, -3]],
"properties": {
"name": "Vertex (9, -3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 97]],
"properties": {
"name": "(4, 97)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [
{
"js": "4 * Math.pow(x - 9, 2) - 3",
"latex": "y = 4(x - 9)^2 - 3"
},
3,
15
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Vertex Form"
]
}
</post_analysis>