x^2=-16y

{ "query": { "display": "$$x^{2}=-16y$$", "symbolab_question": "CONIC#x^{2}=-16y" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,0),p=-4" }, "steps": { "type": "interim", "title": "$$x^{2}=-16y:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:$$and focal length $$|p|=4$$", "input": "x^{2}=-16y", "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 $$x^{2}=-16y\\:$$in the standard form:$${\\quad}4\\left(-4\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}$$", "input": "x^{2}=-16y", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-16y=x^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-16}{4}y=x^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-4\\right)y=x^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-4\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J+7DfqiAJGN1ORFlFeurvbQWgM/Qnh8bwlZVgLtIW15FQXpHylPs0Rersq/WcBN38o9ZLTbv6820yRhHB5ziAqgsumrUE8MI1q8RK7ImYr5ap+53YEFIjmIWpl3qnAYzGk3kCh3oevUunZ7/b0qFKBQr3v3bez3dCmDvWRQb/mMYwA06BWctNYy5sA0btPaS2mkbCg93zvRCzMqKIe6ln76KGM5KjXEoq5HFg4B+mSdM" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-4" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x^{2}}{4(-4)}+0", "displayFormula": "4(-4)y=x^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=4", "displayFormula": "y=4", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(0,-4)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0, "snd": -4 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-4)(y)=x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -45, "xMax": 45, "yMin": -45, "yMax": 45 } }, "showViewLarger": true } } }