(y-3)^2=16(x+5)

{ "query": { "display": "$$\\left(y-3\\right)^{2}=16\\left(x+5\\right)$$", "symbolab_question": "CONIC#(y-3)^{2}=16(x+5)" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-5,3),p=4" }, "steps": { "type": "interim", "title": "$$\\left(y-3\\right)^{2}=16\\left(x+5\\right):\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-5,\\:3\\right),\\:$$and focal length $$|p|=4$$", "input": "\\left(y-3\\right)^{2}=16\\left(x+5\\right)", "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 $$\\left(y-3\\right)^{2}=16\\left(x+5\\right)\\:$$in the standard form:$${\\quad}4\\cdot\\:4\\left(x-\\left(-5\\right)\\right)=\\left(y-3\\right)^{2}$$", "input": "\\left(y-3\\right)^{2}=16\\left(x+5\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "16\\left(x+5\\right)=\\left(y-3\\right)^{2}" }, { "type": "step", "primary": "Factor $$16$$", "result": "16\\left(x+\\frac{80}{16}\\right)=\\left(y-3\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "16\\left(x+5\\right)=\\left(y-3\\right)^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{16}{4}\\left(x+5\\right)=\\left(y-3\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:4\\left(x+5\\right)=\\left(y-3\\right)^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:4\\left(x-\\left(-5\\right)\\right)=\\left(y-3\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7x7c70iKgri0FohdBA3wVDUdEyZcSg/C39X/EGw3HkF4cjlLRK1jUV206qo4+vRN7UQXyIheBU4V7QznAUubIUi9u5h1Ud7PD5KuBB69wYd9L8OVsD2aF1M5xHkeonvWlN3ELcAfoz0vqKpljUqFug0BKzNvThwZxIgCl2Yl3fvOrve3E7cDlwD8G9VYfu6du3xgpIxDNDNEp1Cfe+hNU45sJuiRkp5qUYBSxRmPWUIG4uT2jKtBBZJYZyQX0ofldKeGBIOK4zF36uK2bQJjN9Q==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-5,\\:3\\right),\\:p=4" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\cdot 4(x-(-5))}+3", "displayFormula": "4\\cdot 4(x-(-5))=(y-3)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\cdot 4(x-(-5))}+3", "displayFormula": "4\\cdot 4(x-(-5))=(y-3)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-9", "displayFormula": "x=-9", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-5,3)", "(-1,3)" ], "pointsDecimal": [ { "fst": -5, "snd": 3 }, { "fst": -1, "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\\cdot 4(x-(-5))=(y-3)^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -50, "xMax": 40, "yMin": -42, "yMax": 48 } }, "showViewLarger": true } } }