Question

the equation \\(y = \frac{1}{5}x\\) represents a proportional relationship. explain how you can tell the relationship is proportional from the graph of the equation, and how you can find the constant of proportionality.

Explanation:

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<reasoning>

Identify proportional graph features

Using the Graphing Proportional Relationships knowledge point
\[

$$\begin{aligned} &\text{Graph is a straight line.}\\ &\text{Graph passes through the origin } (0,0). \end{aligned}$$

\]

Identify the constant of proportionality

Using the Constant of Proportionality and Proportional Relationship Equation knowledge points
\[

$$\begin{aligned} &y = kx \implies k = \frac{1}{5}\\ &\text{Constant of proportionality is } \frac{1}{5}. \end{aligned}$$

\]
</reasoning>

<answer>
The graph of the equation \(y = \frac{1}{5}x\) represents a proportional relationship because it is a straight line that passes through the origin \((0,0)\).

The constant of proportionality is the coefficient of \(x\) in the equation \(y = kx\), which is \(\frac{1}{5}\). On the graph, this is represented by the slope of the line, which can be found by taking any point on the line \((x, y)\) other than the origin and calculating the ratio \(\frac{y}{x}\).
</answer>

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"type": "functiongraph",
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Answer:

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<reasoning>

Identify proportional graph features

Using the Graphing Proportional Relationships knowledge point
\[

$$\begin{aligned} &\text{Graph is a straight line.}\\ &\text{Graph passes through the origin } (0,0). \end{aligned}$$

\]

Identify the constant of proportionality

Using the Constant of Proportionality and Proportional Relationship Equation knowledge points
\[

$$\begin{aligned} &y = kx \implies k = \frac{1}{5}\\ &\text{Constant of proportionality is } \frac{1}{5}. \end{aligned}$$

\]
</reasoning>

<answer>
The graph of the equation \(y = \frac{1}{5}x\) represents a proportional relationship because it is a straight line that passes through the origin \((0,0)\).

The constant of proportionality is the coefficient of \(x\) in the equation \(y = kx\), which is \(\frac{1}{5}\). On the graph, this is represented by the slope of the line, which can be found by taking any point on the line \((x, y)\) other than the origin and calculating the ratio \(\frac{y}{x}\).
</answer>

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