4x^2=-y

{ "query": { "display": "$$4x^{2}=-y$$", "symbolab_question": "CONIC#4x^{2}=-y" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,0),p=-\\frac{1}{16}" }, "steps": { "type": "interim", "title": "$$4x^{2}=-y:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:$$and focal length $$|p|=\\frac{1}{16}$$", "input": "4x^{2}=-y", "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 $$4x^{2}=-y\\:$$in the standard form:$${\\quad}4\\left(-\\frac{1}{16}\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}$$", "input": "4x^{2}=-y", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-y=4x^{2}" }, { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{-y}{4}=\\frac{4x^{2}}{4}" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{y}{4}=x^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-\\frac{1}{4}}{4}y=x^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-\\frac{1}{16}\\right)y=x^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-\\frac{1}{16}\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pwKD6ZbvXGMjF+zPZcK4z3c3Rnec5Ic2m/GxjQk9zlteTXjeP8Fry4AS6VRPW/JxnSd9ckJvhaVSw11yD/KmDHfDBptXBowSMDY4Q9A3bJDoEKczQweGAyoyOBw/EP3JFS/tgjVHJg3YOkl4KFP8NxHO0oTnnZveyzJ4AtC1ZGN2WVG1dxuhtwcvc5+As2Tom9FvUgxpVdiPpesJSUWw9X8dB2lu7fHhKm6J3NuxZ/WLzROpyIOFNGyLtHGywk0A" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-\\frac{1}{16}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x^{2}}{4(-\\frac{1}{16})}+0", "displayFormula": "4(-\\frac{1}{16})y=x^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{1}{16}", "displayFormula": "y=\\frac{1}{16}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(0,-\\frac{1}{16})" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0, "snd": -0.0625 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-\\frac{1}{16})(y)=x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -0.703125, "xMax": 0.703125, "yMin": -0.703125, "yMax": 0.703125 } }, "showViewLarger": true } } }