graph the equation.

\(x-3)^2 + (y+2)^2 = 16\

Question

graph the equation.

\(x-3)^2 + (y+2)^2 = 16\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Equation of a Circle", "Graphing Circles" ], "new_concepts": [], "current_concepts": [ "Equation of a Circle", "Graphing Circles" ] } ## Identify the center and radius of the circle $$ \begin{aligned} &(x - h)^2 + (y - k)^2 = r^2 \ &(x - 3)^2 + (y + 2)^2 = 16 \ &h = 3, \quad k = -2 \ &\text{Center: } (3, -2) \ &r^2 = 16 \implies r = 4 \end{aligned} $$ ## Determine key boundary points on the circle $$ \begin{aligned} &\text{Top: } (3, -2 + 4) = (3, 2) \ &\text{Bottom: } (3, -2 - 4) = (3, -6) \ &\text{Right: } (3 + 4, -2) = (7, -2) \ &\text{Left: } (3 - 4, -2) = (-1, -2) \end{aligned} $$ The graph of the equation $(x-3)^2 + (y+2)^2 = 16$ is a circle with: - **Center:** $(3, -2)$ - **Radius:** $4$ - **Key points on the circle:** $(3, 2)$, $(3, -6)$, $(7, -2)$, and $(-1, -2)$ { "elements": [ { "type": "point", "params": [[3, -2]], "properties": { "name": "Center (3, -2)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "circle", "params": [[3, -2], 4], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "fillColor": "rgba(140, 85, 242, 0.15)", "hasInnerPoints": true } }, { "type": "point", "params": [[3, 2]], "properties": { "name": "(3, 2)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [[7, -2]], "properties": { "name": "(7, -2)", "size": 3, "color": "#5583F2", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Graphing Circles" ] }

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Explanation:

Identify the center and radius of the circle

Determine key boundary points on the circle

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