x-y^2=5

{ "query": { "display": "$$x-y^{2}=5$$", "symbolab_question": "CONIC#x-y^{2}=5" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(5,0),p=\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$x-y^{2}=5:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(5,\\:0\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "x-y^{2}=5", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(x-h\\right)=\\left(y-k\\right)^{2}\\:$$ is the standard equation for a right-left facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$x-y^{2}=5\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{4}\\left(x-5\\right)=\\left(y-0\\right)^{2}$$", "input": "x-y^{2}=5", "steps": [ { "type": "step", "primary": "Add $$y^{2}$$ to both sides", "result": "x-y^{2}+y^{2}=5+y^{2}" }, { "type": "step", "primary": "Refine", "result": "x=5+y^{2}" }, { "type": "step", "primary": "Subtract $$5$$ from both sides", "result": "x-5=5+y^{2}-5" }, { "type": "step", "primary": "Refine", "result": "x-5=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{1}{4}\\left(x-5\\right)=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:\\frac{1}{4}\\left(x-5\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J8SR+KK0jtfLvgN0U9EjzKteTXjeP8Fry4AS6VRPW/JxnSd9ckJvhaVSw11yD/KmDHfDBptXBowSMDY4Q9A3bJAIFkbUz7puWZSuakBtmZXz4fNLK3HJvuWoiFNCzngp00BKzNvThwZxIgCl2Yl3fvOrve3E7cDlwD8G9VYfu6duljIs6RL4Pd62fqZKrzS+uk1MaFslp0880EYGMuhB9sRVWM/17S3hH0qEPh3398DMvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(5,\\:0\\right),\\:p=\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\frac{1}{4}(x-5)}+0", "displayFormula": "4\\frac{1}{4}(x-5)=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\frac{1}{4}(x-5)}+0", "displayFormula": "4\\frac{1}{4}(x-5)=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{19}{4}", "displayFormula": "x=\\frac{19}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(5,0)", "(\\frac{21}{4},0)" ], "pointsDecimal": [ { "fst": 5, "snd": 0 }, { "fst": 5.25, "snd": 0 } ], "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}(x-5)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -1.714285714285714, "xMax": 6.857142857142858, "yMin": -4.285714285714286, "yMax": 4.285714285714286 } }, "showViewLarger": true } } }