Question

identify the equation that translates \\(y = \ln(x)\\) five units down.

\\(y = \ln(x-5)\\)
\\(y = \ln(x)+5\\)
\\(y = \ln(x+5)\\)
\\(y = \ln(x)-5\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Function Translation",
"Vertical Translation",
"Logarithmic Transformations"
],
"current_concepts": [
"Function Translation",
"Vertical Translation",
"Logarithmic Transformations"
]
}
</pre_analysis>

<reasoning>

Identify the parent function and transformation rule

\[

$$\begin{aligned} &f(x) = \ln(x)\\ &g(x) = f(x) - c \quad (c > 0 \text{ translates } f(x) \text{ down by } c \text{ units}) \end{aligned}$$

\]

Apply the vertical translation to the function

\[

$$\begin{aligned} &c = 5\\ &y = \ln(x) - 5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \ln(x-5)\)</mcq-option>
<mcq-option>(B) \(y = \ln(x)+5\)</mcq-option>
<mcq-option>(C) \(y = \ln(x+5)\)</mcq-option>
<mcq-correct>(D) \(y = \ln(x)-5\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Translation"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Function Translation",
"Vertical Translation",
"Logarithmic Transformations"
],
"current_concepts": [
"Function Translation",
"Vertical Translation",
"Logarithmic Transformations"
]
}
</pre_analysis>

<reasoning>

Identify the parent function and transformation rule

\[

$$\begin{aligned} &f(x) = \ln(x)\\ &g(x) = f(x) - c \quad (c > 0 \text{ translates } f(x) \text{ down by } c \text{ units}) \end{aligned}$$

\]

Apply the vertical translation to the function

\[

$$\begin{aligned} &c = 5\\ &y = \ln(x) - 5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \ln(x-5)\)</mcq-option>
<mcq-option>(B) \(y = \ln(x)+5\)</mcq-option>
<mcq-option>(C) \(y = \ln(x+5)\)</mcq-option>
<mcq-correct>(D) \(y = \ln(x)-5\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Function Translation"
]
}
</post_analysis>