vertices 3x^2+8y=0

{ "query": { "display": "vertices $$3x^{2}+8y=0$$", "symbolab_question": "CONIC#vertices 3x^{2}+8y=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Maximum}\\:(0,0)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$3x^{2}+8y=0$$", "symbolab_question": "vertexquadratic 3x^{2}+8y=0" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$3x^{2}+8y=0$$", "symbolab_question": "vertexparabola 3x^{2}+8y=0" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$3x^{2}+8y=0$$", "symbolab_question": "vertexform 3x^{2}+8y=0" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$3x^{2}+8y=0$$", "symbolab_question": "vertexzeros 3x^{2}+8y=0" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$3x^{2}+8y=0:{\\quad}$$Maximum $$\\left(0,\\:0\\right)$$", "input": "3x^{2}+8y=0", "steps": [ { "type": "definition", "title": "Parabola equation in polynomial form", "text": "The vertex of an up-down facing parabola of the form $$y=ax^2+bx+c\\:$$is $$x_{v}=-\\frac{b}{2a}$$" }, { "type": "interim", "title": "Rewrite $$3x^{2}+8y=0\\:$$in the form $$y=ax^{2}+bx+c$$", "input": "3x^{2}+8y=0", "result": "y=-\\frac{3x^{2}}{8}", "steps": [ { "type": "interim", "title": "Isolate $$y\\:$$for $$3x^{2}+8y=0$$", "input": "3x^{2}+8y=0", "result": "y=-\\frac{3x^{2}}{8}", "steps": [ { "type": "interim", "title": "Move $$3x^{2}\\:$$to the right side", "input": "3x^{2}+8y=0", "result": "8y=-3x^{2}", "steps": [ { "type": "step", "primary": "Subtract $$3x^{2}$$ from both sides", "result": "3x^{2}+8y-3x^{2}=0-3x^{2}" }, { "type": "step", "primary": "Simplify", "result": "8y=-3x^{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": 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"steps": [ { "type": "step", "primary": "Divide both sides by $$8$$", "result": "\\frac{8y}{8}=\\frac{-3x^{2}}{8}" }, { "type": "step", "primary": "Simplify", "result": "y=-\\frac{3x^{2}}{8}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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2Eq" } }, { "type": "step", "primary": "The parabola parameters are:", "result": "a=-\\frac{3}{8},\\:b=0,\\:c=0" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{0}{2\\left(-\\frac{3}{8}\\right)}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=0" }, { "type": "interim", "title": "Plug in $$x_{v}=0\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=-\\frac{3\\cdot\\:0^{2}}{8}", "result": "y_{v}=0", "steps": [ { "type": "step", "primary": "Simplify", "result": "y_{v}=0" } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(0,\\:0\\right)" }, { "type": "step", "primary": "If $$a<0,\\:$$then the vertex is a maximum value<br/>If $$a>0,\\:$$then the vertex is a minimum value<br/>$$a=-\\frac{3}{8}$$", "result": "\\mathrm{Maximum}\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", 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