vertices x=2y^2

{ "query": { "display": "vertices $$x=2y^{2}$$", "symbolab_question": "CONIC#vertices x=2y^{2}" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "(h,k)=(0,0),p=\\frac{1}{8}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Parabola vertex given $$x=2y^{2}:{\\quad}\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=\\frac{1}{8}$$", "input": "x=2y^{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 $$x=2y^{2}\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{8}\\left(x-0\\right)=\\left(y-0\\right)^{2}$$", "input": "x=2y^{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{x}{2}=\\frac{2y^{2}}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{x}{2}=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{\\frac{1}{2}}{4}x=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:\\frac{1}{8}x=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:\\frac{1}{8}\\left(x-0\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J0XM44gknaY680cf47uq2RbsP/qQDlxeeNYW1ZWJxYAcfPwccPk04ro5gYDy4dxVexA0wxnGmkHWdrtVslPpf+0Q9qdJcuJZcQHTduTfqiU/2Bt1eN7EiQ/vAPIAU2YV9IdEeJB8NSwK2cnf2Bc1WE3bdeLci9PfzmQBgUrYJA7Exrr2x+sFdLUVfadGcXUhNKJlmCzQhfXMb5TwB3CtdANH/gdAmLUqfyKIm7sC2NYG" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=\\frac{1}{8}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\frac{1}{8}x}+0", "displayFormula": "4\\frac{1}{8}x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\frac{1}{8}x}+0", "displayFormula": "4\\frac{1}{8}x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-\\frac{1}{8}", "displayFormula": "x=-\\frac{1}{8}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(\\frac{1}{8},0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0.125, "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}{8}(x)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -1.40625, "xMax": 1.40625, "yMin": -1.40625, "yMax": 1.40625 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }