Question

question 7 (5 points)
use a table of values to graph the function \\(f(x) = \sqrt{x} + 4\\). select the correct graph below.
a

Explanation:

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"concepts_used": [
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"Vertical Translation"
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"new_concepts": [
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<reasoning>

Identify the parent function and its domain

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &y = \sqrt{x}\\ &\text{Domain: } x \ge 0 \end{aligned}$$

\]

Apply the vertical translation

Using the Vertical Translation knowledge point
\[

$$\begin{aligned} &f(x) = \sqrt{x} + 4\\ &\text{Shift: } 4 \text{ units upward}\\ &\text{Starting point: } (0, 4) \end{aligned}$$

\]

Create a table of key values

We select perfect square values for \(x\) to find integer coordinates:

• For \(x = 0\): \(f(0) = \sqrt{0} + 4 = 4 \implies (0, 4)\)
• For \(x = 1\): \(f(1) = \sqrt{1} + 4 = 5 \implies (1, 5)\)
• For \(x = 4\): \(f(4) = \sqrt{4} + 4 = 6 \implies (4, 6)\)
• For \(x = 9\): \(f(9) = \sqrt{9} + 4 = 7 \implies (9, 7)\)

Analyze the given option A

Let's check the graph shown in option A:

• The curve starts at the point \((4, 0)\).
• This corresponds to the function \(y = \sqrt{x - 4}\), not \(f(x) = \sqrt{x} + 4\).
• Therefore, option A is incorrect because it represents a horizontal shift to the right by 4 units instead of a vertical shift up by 4 units.

Determine the correct graph characteristics

The correct graph must:

• Start at \((0, 4)\) on the y-axis.
• Curve upwards and to the right, passing through \((4, 6)\) and \((9, 7)\).

</reasoning>

<answer>
<mcq-option>(A) A curve starting at \((4, 0)\) and passing through \((8, 2)\)</mcq-option>
<mcq-correct>(B) A curve starting at \((0, 4)\) on the y-axis and passing through \((4, 6)\) and \((9, 7)\)</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 4]],
"properties": {
"name": "(0,4)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 6]],
"properties": {
"name": "(4,6)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[9, 7]],
"properties": {
"name": "(9,7)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [
{
"js": "Math.sqrt(x) + 4",
"latex": "\sqrt{x} + 4"
},
0,
10
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Square Root Function Graphing"
]
}
</post_analysis>

Answer:

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"needs_drawing": true,
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<reasoning>

Identify the parent function and its domain

Using the Function Transformations knowledge point
\[

$$\begin{aligned} &y = \sqrt{x}\\ &\text{Domain: } x \ge 0 \end{aligned}$$

\]

Apply the vertical translation

Using the Vertical Translation knowledge point
\[

$$\begin{aligned} &f(x) = \sqrt{x} + 4\\ &\text{Shift: } 4 \text{ units upward}\\ &\text{Starting point: } (0, 4) \end{aligned}$$

\]

Create a table of key values

We select perfect square values for \(x\) to find integer coordinates:

• For \(x = 0\): \(f(0) = \sqrt{0} + 4 = 4 \implies (0, 4)\)
• For \(x = 1\): \(f(1) = \sqrt{1} + 4 = 5 \implies (1, 5)\)
• For \(x = 4\): \(f(4) = \sqrt{4} + 4 = 6 \implies (4, 6)\)
• For \(x = 9\): \(f(9) = \sqrt{9} + 4 = 7 \implies (9, 7)\)

Analyze the given option A

Let's check the graph shown in option A:

• The curve starts at the point \((4, 0)\).
• This corresponds to the function \(y = \sqrt{x - 4}\), not \(f(x) = \sqrt{x} + 4\).
• Therefore, option A is incorrect because it represents a horizontal shift to the right by 4 units instead of a vertical shift up by 4 units.

Determine the correct graph characteristics

The correct graph must:

• Start at \((0, 4)\) on the y-axis.
• Curve upwards and to the right, passing through \((4, 6)\) and \((9, 7)\).

</reasoning>

<answer>
<mcq-option>(A) A curve starting at \((4, 0)\) and passing through \((8, 2)\)</mcq-option>
<mcq-correct>(B) A curve starting at \((0, 4)\) on the y-axis and passing through \((4, 6)\) and \((9, 7)\)</mcq-correct>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 4]],
"properties": {
"name": "(0,4)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 6]],
"properties": {
"name": "(4,6)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[9, 7]],
"properties": {
"name": "(9,7)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [
{
"js": "Math.sqrt(x) + 4",
"latex": "\sqrt{x} + 4"
},
0,
10
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Square Root Function Graphing"
]
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