x^2=y-4

{ "query": { "display": "$$x^{2}=y-4$$", "symbolab_question": "CONIC#x^{2}=y-4" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,4),p=\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$x^{2}=y-4:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:4\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "x^{2}=y-4", "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 $$x^{2}=y-4\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{4}\\left(y-4\\right)=\\left(x-0\\right)^{2}$$", "input": "x^{2}=y-4", "steps": [ { "type": "step", "primary": "Switch sides", "result": "y-4=x^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{1}{4}\\left(y-4\\right)=x^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:\\frac{1}{4}\\left(y-4\\right)=\\left(x-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J3bgD+uUQhZ88+P2K/9mc4FeTXjeP8Fry4AS6VRPW/JxnSd9ckJvhaVSw11yD/KmDHfDBptXBowSMDY4Q9A3bJAIFkbUz7puWZSuakBtmZXz5ns+4AaWCw5ma8AU5QRa4EBKzNvThwZxIgCl2Yl3fvOrve3E7cDlwD8G9VYfu6duMU/JeSXTroiKcrJGqvNBIE1MaFslp0880EYGMuhB9sRVWM/17S3hH0qEPh3398DMvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:4\\right),\\:p=\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x^{2}}{4\\frac{1}{4}}+4", "displayFormula": "4\\frac{1}{4}(y-4)=x^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{15}{4}", "displayFormula": "y=\\frac{15}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,4)", "(0,\\frac{17}{4})" ], "pointsDecimal": [ { "fst": 0, "snd": 4 }, { "fst": 0, "snd": 4.25 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4\\cdot \\frac{1}{4}(y-4)=x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -3.5714285714285716, "xMax": 3.5714285714285716, "yMin": -1.4285714285714288, "yMax": 5.714285714285714 } }, "showViewLarger": true } } }