Question

compute the slopes for the two given functions and determine whether the functions are parallel or perpendicular.
i) \\(y = -5x + 20\\)
(ii) graph function

the given functions are parallel.
the given functions are neither parallel nor perpendicular.

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": [],
"current_concepts": [
"Slope of a Linear Function",
"Parallel and Perpendicular Lines",
"Slope-Intercept Form"
]
}
</pre_analysis>

<reasoning>

Find the slope of the first function

\[

$$\begin{aligned} &y = -5x + 20\\ &m_1 = -5 \end{aligned}$$

\]

Find the slope of the second function from the graph

\[

$$\begin{aligned} &\text{Points on the line: } (0, 4) \text{ and } (-2, 0)\\ &m_2 = \frac{4 - 0}{0 - (-2)} = \frac{4}{2} = 2 \end{aligned}$$

\]

Compare the slopes to determine the relationship

\[

$$\begin{aligned} &m_1 = -5,\quad m_2 = 2\\ &m_1 e m_2 \quad (\text{not parallel})\\ &m_1 \cdot m_2 = -10 e -1 \quad (\text{not perpendicular}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>The given functions are parallel.</mcq-option>
<mcq-correct>The given functions are neither parallel nor perpendicular.</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Parallel and Perpendicular Lines"
]
}
</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": [],
"current_concepts": [
"Slope of a Linear Function",
"Parallel and Perpendicular Lines",
"Slope-Intercept Form"
]
}
</pre_analysis>

<reasoning>

Find the slope of the first function

\[

$$\begin{aligned} &y = -5x + 20\\ &m_1 = -5 \end{aligned}$$

\]

Find the slope of the second function from the graph

\[

$$\begin{aligned} &\text{Points on the line: } (0, 4) \text{ and } (-2, 0)\\ &m_2 = \frac{4 - 0}{0 - (-2)} = \frac{4}{2} = 2 \end{aligned}$$

\]

Compare the slopes to determine the relationship

\[

$$\begin{aligned} &m_1 = -5,\quad m_2 = 2\\ &m_1 e m_2 \quad (\text{not parallel})\\ &m_1 \cdot m_2 = -10 e -1 \quad (\text{not perpendicular}) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>The given functions are parallel.</mcq-option>
<mcq-correct>The given functions are neither parallel nor perpendicular.</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Parallel and Perpendicular Lines"
]
}
</post_analysis>