Question

question 6 points 2

which function has a greater rate of change?

\\(y = 2 - 0.9x\\)

\\(2y = 4x + 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": [
"Slope of a Linear Function",
"Slope-Intercept Form"
],
"new_concepts": [
"Rate of Change Comparison"
],
"current_concepts": [
"Slope of a Linear Function",
"Slope-Intercept Form",
"Rate of Change Comparison"
]
}
</pre_analysis>

<reasoning>

Find the rate of change of the first function

\[

$$\begin{aligned} &y = 2 - 0.9x \\ &y = -0.9x + 2 \\ &m_1 = -0.9 \\ &\text{Rate of change (magnitude)} = |-0.9| = 0.9 \end{aligned}$$

\]

Find the rate of change of the second function

\[

$$\begin{aligned} &2y = 4x + 5 \\ &y = 2x + 2.5 \\ &m_2 = 2 \\ &\text{Rate of change (magnitude)} = |2| = 2 \end{aligned}$$

\]

Compare the rates of change

\[

$$\begin{aligned} &2 > 0.9 \\ &\text{The function } 2y = 4x + 5 \text{ has a greater rate of change.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(y = 2 - 0.9x\)</mcq-option>
<mcq-correct>\(2y = 4x + 5\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Slope of a Linear Function"
]
}
</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": [
"Slope of a Linear Function",
"Slope-Intercept Form"
],
"new_concepts": [
"Rate of Change Comparison"
],
"current_concepts": [
"Slope of a Linear Function",
"Slope-Intercept Form",
"Rate of Change Comparison"
]
}
</pre_analysis>

<reasoning>

Find the rate of change of the first function

\[

$$\begin{aligned} &y = 2 - 0.9x \\ &y = -0.9x + 2 \\ &m_1 = -0.9 \\ &\text{Rate of change (magnitude)} = |-0.9| = 0.9 \end{aligned}$$

\]

Find the rate of change of the second function

\[

$$\begin{aligned} &2y = 4x + 5 \\ &y = 2x + 2.5 \\ &m_2 = 2 \\ &\text{Rate of change (magnitude)} = |2| = 2 \end{aligned}$$

\]

Compare the rates of change

\[

$$\begin{aligned} &2 > 0.9 \\ &\text{The function } 2y = 4x + 5 \text{ has a greater rate of change.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(y = 2 - 0.9x\)</mcq-option>
<mcq-correct>\(2y = 4x + 5\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Slope of a Linear Function"
]
}
</post_analysis>