which graph represents $y - 1 = 2(x - 2)$?

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Linear Functions", "Slope-Intercept Form" ], "new_concepts": [ "Point-Slope Form" ], "current_concepts": [ "Linear Functions", "Slope-Intercept Form", "Point-Slope Form" ] } ## Convert the equation to slope-intercept form $$ \begin{aligned} y - 1 &= 2(x - 2) \ y - 1 &= 2x - 4 \ y &= 2x - 3 \end{aligned} $$ ## Identify key features of the line $$ \begin{aligned} \text{Slope } (m) &= 2 \ y\text{-intercept } (b) &= -3 \implies (0, -3) \ x\text{-intercept } (x_0) &\implies 0 = 2x - 3 \implies x = 1.5 \implies (1.5, 0) \end{aligned} $$ ## Match features with the first graph $$ \begin{aligned} \text{Graph } y\text{-intercept} &= (0, -3) \ \text{Graph } x\text{-intercept} &= (1.5, 0) \ \text{Graph point } (2, 1) &\implies 1 - 1 = 2(2 - 2) \implies 0 = 0 \quad (\text{True}) \end{aligned} $$

(A) The first graph, which passes through the $y$-intercept $(0, -3)$ and has a slope of $2$.

(B) The second graph, which does not pass through $(0, -3)$.

{ "elements": [ { "type": "point", "params": [[2, 1]], "properties": { "name": "(2, 1)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, -3]], "properties": { "name": "(0, -3)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "functiongraph", "params": [{"js": "2*x - 3", "latex": "2x - 3"}, -6, 6], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Linear Functions" ] }

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Question

Explanation:

Convert the equation to slope-intercept form

Identify key features of the line

Match features with the first graph

Answer: