x=3y^2

{ "query": { "display": "$$x=3y^{2}$$", "symbolab_question": "CONIC#x=3y^{2}" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,0),p=\\frac{1}{12}" }, "steps": { "type": "interim", "title": "$$x=3y^{2}:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:$$and focal length $$|p|=\\frac{1}{12}$$", "input": "x=3y^{2}", "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 $$x=3y^{2}\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{12}\\left(x-0\\right)=\\left(y-0\\right)^{2}$$", "input": "x=3y^{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{x}{3}=\\frac{3y^{2}}{3}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x}{3}=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{\\frac{1}{3}}{4}x=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:\\frac{1}{12}x=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:\\frac{1}{12}\\left(x-0\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J+ICgXJ4Cd9yRyqYlJnfcIPsP/qQDlxeeNYW1ZWJxYAcfPwccPk04ro5gYDy4dxVexA0wxnGmkHWdrtVslPpf+0Q9qdJcuJZcQHTduTfqiU/aMukGpvKVutL4X5j8dyfMEBKzNvThwZxIgCl2Yl3fvOrve3E7cDlwD8G9VYfu6duRIMBF+ohPGZDN/D3oSP19IE5KqSizxQcz0mXfJLnLF/KSq/BachEG89jYXgLuVVqJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=\\frac{1}{12}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\frac{1}{12}x}+0", "displayFormula": "4\\frac{1}{12}x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\frac{1}{12}x}+0", "displayFormula": "4\\frac{1}{12}x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-\\frac{1}{12}", "displayFormula": "x=-\\frac{1}{12}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(\\frac{1}{12},0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0.08333333333333333, "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}{12}(x)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -0.9375, "xMax": 0.9375, "yMin": -0.9375, "yMax": 0.9375 } }, "showViewLarger": true } } }