y-x>-x^2-1

{ "query": { "display": "$$y-x>-x^{2}-1$$", "symbolab_question": "CONIC#y-x>-x^{2}-1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(\\frac{1}{2},-\\frac{3}{4}),p=-\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$y-x>-x^{2}-1:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(\\frac{1}{2},\\:-\\frac{3}{4}\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "y-x>-x^{2}-1", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(y-k\\right)=\\left(x-h\\right)^{2}\\:$$ is the standard equation for an up-down facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$y-x>-x^{2}-1\\:$$in the standard form", "input": "y-x>-x^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "y=-x^{2}+x-1" }, { "type": "interim", "title": "Complete the square $$x-x^{2}-1:{\\quad}-\\left(x-\\frac{1}{2}\\right)^{2}-\\frac{3}{4}$$", "input": "x-x^{2}-1", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c$$", "result": "-x^{2}+x-1" }, { "type": "step", "primary": "Write $$-x^{2}+x-1\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-1$$" ], "result": "-\\left(x^{2}-x+1\\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(x^{2}-x+1+\\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": [ "$$x^{2}-1x+\\left(-\\frac{1}{2}\\right)^{2}=\\left(x-\\frac{1}{2}\\right)^{2}$$", "Complete the square" ], "result": "-\\left(\\left(x-\\frac{1}{2}\\right)^{2}+1-\\left(-\\frac{1}{2}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\left(x-\\frac{1}{2}\\right)^{2}-\\frac{3}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=-\\left(x-\\frac{1}{2}\\right)^{2}-\\frac{3}{4}" }, { "type": "step", "primary": "Add $$\\frac{3}{4}$$ to both sides", "result": "y+\\frac{3}{4}=-\\left(x-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-1$$", "result": "-\\left(y+\\frac{3}{4}\\right)=\\left(x-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-\\frac{1}{4}\\right)\\left(y-\\left(-\\frac{3}{4}\\right)\\right)=\\left(x-\\frac{1}{2}\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-\\frac{1}{4}\\right)\\left(y-\\left(-\\frac{3}{4}\\right)\\right)=\\left(x-\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(\\frac{1}{2},\\:-\\frac{3}{4}\\right),\\:p=-\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-\\frac{1}{2})^{2}}{4(-\\frac{1}{4})}-\\frac{3}{4}", "displayFormula": "4(-\\frac{1}{4})(y-(-\\frac{3}{4}))=(x-\\frac{1}{2})^{2}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{1}{2}", "displayFormula": "y=-\\frac{1}{2}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(\\frac{1}{2},-\\frac{3}{4})", "(\\frac{1}{2},-1)" ], "pointsDecimal": [ { "fst": 0.5, "snd": -0.75 }, { "fst": 0.5, 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