vertices 3y^2=24x

{ "query": { "display": "vertices $$3y^{2}=24x$$", "symbolab_question": "CONIC#vertices 3y^{2}=24x" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "(h,k)=(0,0),p=2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Parabola vertex given $$3y^{2}=24x:{\\quad}\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=2$$", "input": "3y^{2}=24x", "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 $$3y^{2}=24x\\:$$in the standard form:$${\\quad}4\\cdot\\:2\\left(x-0\\right)=\\left(y-0\\right)^{2}$$", "input": "3y^{2}=24x", "steps": [ { "type": "step", "primary": "Switch sides", "result": "24x=3y^{2}" }, { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{24x}{3}=\\frac{3y^{2}}{3}" }, { "type": "step", "primary": "Simplify", "result": "8x=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{8}{4}x=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:2x=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:2\\left(x-0\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/fpQlIGst70YQiFj6wvjJD9pjBYVXGxlT2qGsMxUtIAWgM/Qnh8bwlZVgLtIW15FQXpHylPs0Rersq/WcBN38o9ZLTbv6820yRhHB5ziAqjxDO8yXE1Q2E64JiTdc5CNwnx5/ddVgLKa5Ler5RfIoYEFMST8lDZxn1Yq5HMKVTuO5g1V2JtJrnbMKBfmVthppD9W7kZ26iD0b2AR42i6e3u+x6S+J00Ipg6ECvPsq0JTOtQuxiLY1bb+sl5tgE8F" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=2" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\cdot 2x}+0", "displayFormula": "4\\cdot 2x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\cdot 2x}+0", "displayFormula": "4\\cdot 2x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-2", "displayFormula": "x=-2", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(2,0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 2, "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 2(x)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -22.5, "xMax": 22.5, "yMin": -22.5, "yMax": 22.5 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }