12y+x^2=0

{ "query": { "display": "$$12y+x^{2}=0$$", "symbolab_question": "CONIC#12y+x^{2}=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,0),p=-3" }, "steps": { "type": "interim", "title": "$$12y+x^{2}=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:$$and focal length $$|p|=3$$", "input": "12y+x^{2}=0", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(y-k\\right)=\\left(x-h\\right)^{2}\\:$$ is the standard equation for an up-down facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$12y+x^{2}=0\\:$$in the standard form:$${\\quad}4\\left(-3\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}$$", "input": "12y+x^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$x^{2}$$ from both sides", "result": "12y+x^{2}-x^{2}=0-x^{2}" }, { "type": "step", "primary": "Refine", "result": "12y=-x^{2}" }, { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{12y}{-1}=\\frac{-x^{2}}{-1}" }, { "type": "step", "primary": "Simplify", "result": "-12y=x^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-12}{4}y=x^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-3\\right)y=x^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-3\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L1hU6UYRNlySJ7LfHaV09Rnz2PGleSdJAw6jQ13MnVgjGsBIMn028UBVvff51AwIUo+0skB3JhOoVsRyoC4GXjh2no6HYikYL/uXZaE/9Y/apXczCuSTG7CwPMGE80Aa0R2ezmQjS5W1eo+O0ihzAHql8XXPq6bNQlMm+36iNhnKjiytAWaK9jbpnb3r31dEBBvyJMgy3QgMp5ubn9U5SYHrgyXxvQqLRZoailGYyyYfT9nLmYg2mxBVBWxtPcsS" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-3" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x^{2}}{4(-3)}+0", "displayFormula": "4(-3)y=x^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=3", "displayFormula": "y=3", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(0,-3)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0, "snd": -3 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-3)(y)=x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -33.75, "xMax": 33.75, "yMin": -33.75, "yMax": 33.75 } }, "showViewLarger": true } } }