x=y-y^2

{ "query": { "display": "$$x=y-y^{2}$$", "symbolab_question": "CONIC#x=y-y^{2}" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(\\frac{1}{4},\\frac{1}{2}),p=-\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$x=y-y^{2}:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(\\frac{1}{4},\\:\\frac{1}{2}\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "x=y-y^{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 $$x=y-y^{2}\\:$$in the standard form", "input": "x=y-y^{2}", "steps": [ { "type": "interim", "title": "Complete the square $$y-y^{2}:{\\quad}-\\left(y-\\frac{1}{2}\\right)^{2}+\\frac{1}{4}$$", "input": "y-y^{2}", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c$$", "result": "-y^{2}+y" }, { "type": "step", "primary": "Write $$-y^{2}+y\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-1$$" ], "result": "-\\left(y^{2}-y\\right)" }, { "type": "interim", "title": "$$2a=-1{\\quad:\\quad}a=-\\frac{1}{2}$$", "input": "2a=-1", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=-1", "result": "a=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{-1}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(-\\frac{1}{2}\\right)^{2}\\:$$", "result": "-\\left(y^{2}-y+\\left(-\\frac{1}{2}\\right)^{2}-\\left(-\\frac{1}{2}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$y^{2}-1y+\\left(-\\frac{1}{2}\\right)^{2}=\\left(y-\\frac{1}{2}\\right)^{2}$$", "Complete the square" ], "result": "-\\left(\\left(y-\\frac{1}{2}\\right)^{2}-\\left(-\\frac{1}{2}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\left(y-\\frac{1}{2}\\right)^{2}+\\frac{1}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "x=-\\left(y-\\frac{1}{2}\\right)^{2}+\\frac{1}{4}" }, { "type": "step", "primary": "Subtract $$\\frac{1}{4}$$ from both sides", "result": "x-\\frac{1}{4}=-\\left(y-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-1$$", "result": "-\\left(x-\\frac{1}{4}\\right)=\\left(y-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-\\frac{1}{4}\\right)\\left(x-\\frac{1}{4}\\right)=\\left(y-\\frac{1}{2}\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-\\frac{1}{4}\\right)\\left(x-\\frac{1}{4}\\right)=\\left(y-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(\\frac{1}{4},\\:\\frac{1}{2}\\right),\\:p=-\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-\\frac{1}{4})(x-\\frac{1}{4})}+\\frac{1}{2}", "displayFormula": "4(-\\frac{1}{4})(x-\\frac{1}{4})=(y-\\frac{1}{2})^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-\\frac{1}{4})(x-\\frac{1}{4})}+\\frac{1}{2}", "displayFormula": "4(-\\frac{1}{4})(x-\\frac{1}{4})=(y-\\frac{1}{2})^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{1}{2}", "displayFormula": "x=\\frac{1}{2}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": 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