y^2-2x=0

{ "query": { "display": "$$y^{2}-2x=0$$", "symbolab_question": "CONIC#y^{2}-2x=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,0),p=\\frac{1}{2}" }, "steps": { "type": "interim", "title": "$$y^{2}-2x=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:$$and focal length $$|p|=\\frac{1}{2}$$", "input": "y^{2}-2x=0", "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 $$y^{2}-2x=0\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{2}\\left(x-0\\right)=\\left(y-0\\right)^{2}$$", "input": "y^{2}-2x=0", "steps": [ { "type": "step", "primary": "Subtract $$y^{2}$$ from both sides", "result": "y^{2}-2x-y^{2}=0-y^{2}" }, { "type": "step", "primary": "Refine", "result": "-2x=-y^{2}" }, { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-2x}{-1}=\\frac{-y^{2}}{-1}" }, { "type": "step", "primary": "Simplify", "result": "2x=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{2}{4}x=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:\\frac{1}{2}x=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:\\frac{1}{2}\\left(x-0\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aRgRQFW0MqhVT4GbzWEc4vSGbcwDMMI2L2FXQ9eIOWkWgM/Qnh8bwlZVgLtIW15FQXpHylPs0Rersq/WcBN38o9ZLTbv6820yRhHB5ziAqjmMk3O3+UQ+jFPL/S5k05QigsLtEBStv7Om6emW/m3iUfGrS3SgBFgWE4O7Y/3J2G9dDZkXCs5+Cxg1YNvJNPUlNmZRKA8miaz9fhtOvVFVM71x0+nbg2xuiMX43R25FTQKUU/hwR/NhRlKnzEcm3cialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=\\frac{1}{2}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\frac{1}{2}x}+0", "displayFormula": "4\\frac{1}{2}x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\frac{1}{2}x}+0", "displayFormula": "4\\frac{1}{2}x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-\\frac{1}{2}", "displayFormula": "x=-\\frac{1}{2}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(\\frac{1}{2},0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0.5, "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}{2}(x)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -5.625, "xMax": 5.625, "yMin": -5.625, "yMax": 5.625 } }, "showViewLarger": true } } }