graph each equation.

9) \frac{x^2}{4} + \frac{y^2}{9} = 1

Question

graph each equation.

9) \frac{x^2}{4} + \frac{y^2}{9} = 1

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Ellipse Graphing", "Conic Sections" ], "new_concepts": [], "current_concepts": [ "Ellipse Graphing", "Conic Sections" ] } ## Identify the key parameters of the ellipse $$ \begin{aligned} &\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{x^2}{4} + \frac{y^2}{9} = 1\ &\text{Center: } (0, 0)\ &a^2 = 4 \implies a = 2 \quad (\text{horizontal semi-axis})\ &b^2 = 9 \implies b = 3 \quad (\text{vertical semi-axis}) \end{aligned} $$ ## Determine the key boundary points $$ \begin{aligned} &\text{Vertices on x-axis: } (2, 0), (-2, 0)\ &\text{Vertices on y-axis: } (0, 3), (0, -3) \end{aligned} $$ The graph of the equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ is a vertical ellipse centered at the origin $(0,0)$ with horizontal vertices at $(2, 0)$ and $(-2, 0)$, and vertical vertices at $(0, 3)$ and $(0, -3)$. { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "Center (0,0)", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[2, 0]], "properties": { "name": "(2,0)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[-2, 0]], "properties": { "name": "(-2,0)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, 3]], "properties": { "name": "(0,3)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, -3]], "properties": { "name": "(0,-3)", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "ellipse", "params": [[0, 0], 2, 3], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "fillColor": "rgba(140, 85, 242, 0.1)" } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Ellipse Graphing" ] }

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Identify the key parameters of the ellipse

Determine the key boundary points

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