Question

which of the following is the graph of this square root function?
\\f(x) = \sqrt{x - 2} + 2\\

Explanation:

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<reasoning>

Find the domain of the function

Using the Domain of Radical Functions knowledge point
\[

$$\begin{aligned} x - 2 &\ge 0 \\ x &\ge 2 \end{aligned}$$

\]

Identify the starting point

Using the Radical Function Transformations and Square Root Graphs knowledge points
\[

$$\begin{aligned} f(2) &= \sqrt{2 - 2} + 2 \\ &= 0 + 2 \\ &= 2 \end{aligned}$$

\]
The starting point of the graph is \((2, 2)\).

Evaluate an additional point

Using the Square Root Graphs knowledge point
\[

$$\begin{aligned} f(6) &= \sqrt{6 - 2} + 2 \\ &= \sqrt{4} + 2 \\ &= 2 + 2 \\ &= 4 \end{aligned}$$

\]
The graph passes through the point \((6, 4)\).

Match with the given graphs

Analyzing the three options:

• First graph: Starts at \((-2, -2)\).
• Second graph: Starts at \((-1, 2)\).
• Third graph: Starts at \((2, 2)\) and passes through \((6, 4)\).

</reasoning>

<answer>
<mcq-option>(A) Graph starting at \((-2, -2)\)</mcq-option>
<mcq-option>(B) Graph starting at \((-1, 2)\)</mcq-option>
<mcq-correct>(C) Graph starting at \((2, 2)\)</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "Math.sqrt(x - 2) + 2",
"latex": "\sqrt{x - 2} + 2"
},
2,
10
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
},
{
"type": "point",
"params": [[2, 2]],
"properties": {
"name": "(2,2)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 4]],
"properties": {
"name": "(6,4)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
}
]
}
</plot>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Square Root Graphs"
]
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Answer:

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</pre_analysis>

<reasoning>

Find the domain of the function

Using the Domain of Radical Functions knowledge point
\[

$$\begin{aligned} x - 2 &\ge 0 \\ x &\ge 2 \end{aligned}$$

\]

Identify the starting point

Using the Radical Function Transformations and Square Root Graphs knowledge points
\[

$$\begin{aligned} f(2) &= \sqrt{2 - 2} + 2 \\ &= 0 + 2 \\ &= 2 \end{aligned}$$

\]
The starting point of the graph is \((2, 2)\).

Evaluate an additional point

Using the Square Root Graphs knowledge point
\[

$$\begin{aligned} f(6) &= \sqrt{6 - 2} + 2 \\ &= \sqrt{4} + 2 \\ &= 2 + 2 \\ &= 4 \end{aligned}$$

\]
The graph passes through the point \((6, 4)\).

Match with the given graphs

Analyzing the three options:

• First graph: Starts at \((-2, -2)\).
• Second graph: Starts at \((-1, 2)\).
• Third graph: Starts at \((2, 2)\) and passes through \((6, 4)\).

</reasoning>

<answer>
<mcq-option>(A) Graph starting at \((-2, -2)\)</mcq-option>
<mcq-option>(B) Graph starting at \((-1, 2)\)</mcq-option>
<mcq-correct>(C) Graph starting at \((2, 2)\)</mcq-correct>
</answer>

<plot>
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"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "Math.sqrt(x - 2) + 2",
"latex": "\sqrt{x - 2} + 2"
},
2,
10
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
},
{
"type": "point",
"params": [[2, 2]],
"properties": {
"name": "(2,2)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 4]],
"properties": {
"name": "(6,4)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
}
]
}
</plot>

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"question_type": "Multiple Choice",
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"Square Root Graphs"
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