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.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Constant of Proportionality", "Proportional Relationship Equation", "Graphing Proportional Relationships" ], "new_concepts": [], "current_concepts": [ "Constant of Proportionality", "Proportional Relationship Equation", "Graphing Proportional Relationships" ] } ## 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 *Proportional Relationship Equation* and *Constant of Proportionality* knowledge points $$ \begin{aligned} &y = kx \implies y = \frac{1}{5}x\ &k = \frac{1}{5} \end{aligned} $$ A graph represents a proportional relationship if it is a straight line that passes through the origin $(0,0)$. For the equation $y = \frac{1}{5}x$, the constant of proportionality is the coefficient of $x$, which is $\frac{1}{5}$. { "elements": [ { "type": "functiongraph", "params": [ { "js": "0.2*x", "latex": "\frac{1}{5}x" }, -2, 10 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "y = 1/5 x", "withLabel": true } }, { "type": "point", "params": [ [0, 0] ], "properties": { "name": "(0,0)", "size": 4, "color": "#F2557F", "strokeColor": "#F2557F", "fillColor": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [5, 1] ], "properties": { "name": "(5,1)", "size": 4, "color": "#5583F2", "strokeColor": "#5583F2", "fillColor": "#5583F2", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Graphing Proportional Relationships" ] }

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Identify proportional graph features

Identify the constant of proportionality

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