Question

graph the function.

\\f(x)=\sqrt{x}+5\\

plot four points on the graph of the function: the leftmost point and three additional points. then click on the graph-a-function button.

Explanation:

<pre_analysis>
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"step_cot",
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"current_concepts": [
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"Function Transformations"
]
}
</pre_analysis>

<reasoning>

Determine the domain and leftmost point

\[

$$\begin{aligned} x + 5 &\ge 0 \implies x \ge -5 \\ f(-5) &= \sqrt{-5 + 5} = 0 \\ \text{Leftmost point: } &(-5, 0) \end{aligned}$$

\]

Calculate three additional integer points

\[

$$\begin{aligned} x = -4 &\implies f(-4) = \sqrt{-4 + 5} = 1 \implies (-4, 1) \\ x = -1 &\implies f(-1) = \sqrt{-1 + 5} = 2 \implies (-1, 2) \\ x = 4 &\implies f(4) = \sqrt{4 + 5} = 3 \implies (4, 3) \end{aligned}$$

\]
</reasoning>

<answer>
The four points to plot on the graph of \(f(x) = \sqrt{x+5}\) are:

• Leftmost point: \((-5, 0)\)
• Additional point 1: \((-4, 1)\)
• Additional point 2: \((-1, 2)\)
• Additional point 3: \((4, 3)\)

</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-5, 0]],
"properties": {
"name": "(-5,0)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-4, 1]],
"properties": {
"name": "(-4,1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-1, 2]],
"properties": {
"name": "(-1,2)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 3]],
"properties": {
"name": "(4,3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [
{
"js": "Math.sqrt(x + 5)",
"latex": "\sqrt{x+5}"
},
-5,
13
],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.2, 0.4, 0.6, 0.8, 1.2]
}
</plot>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Square Root Function Graphing"
]
}
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Answer:

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"quick"
],
"needs_drawing": true,
"concepts_used": [],
"new_concepts": [
"Square Root Function Graphing",
"Domain of Radical Functions"
],
"current_concepts": [
"Square Root Function Graphing",
"Domain of Radical Functions",
"Function Transformations"
]
}
</pre_analysis>

<reasoning>

Determine the domain and leftmost point

\[

$$\begin{aligned} x + 5 &\ge 0 \implies x \ge -5 \\ f(-5) &= \sqrt{-5 + 5} = 0 \\ \text{Leftmost point: } &(-5, 0) \end{aligned}$$

\]

Calculate three additional integer points

\[

$$\begin{aligned} x = -4 &\implies f(-4) = \sqrt{-4 + 5} = 1 \implies (-4, 1) \\ x = -1 &\implies f(-1) = \sqrt{-1 + 5} = 2 \implies (-1, 2) \\ x = 4 &\implies f(4) = \sqrt{4 + 5} = 3 \implies (4, 3) \end{aligned}$$

\]
</reasoning>

<answer>
The four points to plot on the graph of \(f(x) = \sqrt{x+5}\) are:

• Leftmost point: \((-5, 0)\)
• Additional point 1: \((-4, 1)\)
• Additional point 2: \((-1, 2)\)
• Additional point 3: \((4, 3)\)

</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-5, 0]],
"properties": {
"name": "(-5,0)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-4, 1]],
"properties": {
"name": "(-4,1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-1, 2]],
"properties": {
"name": "(-1,2)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 3]],
"properties": {
"name": "(4,3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [
{
"js": "Math.sqrt(x + 5)",
"latex": "\sqrt{x+5}"
},
-5,
13
],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.2, 0.4, 0.6, 0.8, 1.2]
}
</plot>

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"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Square Root Function Graphing"
]
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