{ "query": { "display": "$$\\left(x-3\\right)^{2}=8\\left(y-2\\right)$$", "symbolab_question": "CONIC#(x-3)^{2}=8(y-2)" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(3,2),p=2" }, "steps": { "type": "interim", "title": "$$\\left(x-3\\right)^{2}=8\\left(y-2\\right):\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(3,\\:2\\right),\\:$$and focal length $$|p|=2$$", "input": "\\left(x-3\\right)^{2}=8\\left(y-2\\right)", "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 $$\\left(x-3\\right)^{2}=8\\left(y-2\\right)\\:$$in the standard form:$${\\quad}4\\cdot\\:2\\left(y-2\\right)=\\left(x-3\\right)^{2}$$", "input": "\\left(x-3\\right)^{2}=8\\left(y-2\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "8\\left(y-2\\right)=\\left(x-3\\right)^{2}" }, { "type": "step", "primary": "Factor $$8$$", "result": "8\\left(y+\\frac{-16}{8}\\right)=\\left(x-3\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "8\\left(y-2\\right)=\\left(x-3\\right)^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{8}{4}\\left(y-2\\right)=\\left(x-3\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:2\\left(y-2\\right)=\\left(x-3\\right)^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:2\\left(y-2\\right)=\\left(x-3\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7x7c70iKgri0FohdBA3wVDeSV/4dmSIf2wz34h7AuPmOTdaV09PMxEKZ9FieghTFwIHjjrlQK96/oL7qEXVfedeEfpojVh0cP5vSMCDPkeM10BNumxmVU/tlY6yuoAdGPDQX0ZKJOCf04qYjohD4YtP8//6/nV5O4fb8Xgwi7maqIP2TH9VLmoizBQwdUXcYEkjY/D+BfdIj+0PqU+nm3Xwoz5I5Xp2hOBZ2vMDvjy2RpGwoPd870QszKiiHupZ++ihjOSo1xKKuRxYOAfpknTA==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(3,\\:2\\right),\\:p=2" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-3)^{2}}{4\\cdot 2}+2", "displayFormula": "4\\cdot 2(y-2)=(x-3)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=0", "displayFormula": "y=0", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(3,2)", "(3,4)" ], "pointsDecimal": [ { "fst": 3, "snd": 2 }, { "fst": 3, "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\\cdot 2(y-2)=(x-3)^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -19.5, "xMax": 25.5, "yMin": -20.5, "yMax": 24.5 } }, "showViewLarger": true } } }