y>-x^2-3x+1

{ "query": { "display": "$$y>-x^{2}-3x+1$$", "symbolab_question": "CONIC#y>-x^{2}-3x+1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-\\frac{3}{2},\\frac{13}{4}),p=-\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$y>-x^{2}-3x+1:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-\\frac{3}{2},\\:\\frac{13}{4}\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "y>-x^{2}-3x+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 $$y>-x^{2}-3x+1\\:$$in the standard form", "input": "y>-x^{2}-3x+1", "steps": [ { "type": "interim", "title": "Complete the square $$-x^{2}-3x+1:{\\quad}-\\left(x+\\frac{3}{2}\\right)^{2}+\\frac{13}{4}$$", "input": "-x^{2}-3x+1", "steps": [ { "type": "step", "primary": "Write $$-x^{2}-3x+1\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-1$$" ], "result": "-\\left(x^{2}+3x-1\\right)" }, { "type": "interim", "title": "$$2a=3{\\quad:\\quad}a=\\frac{3}{2}$$", "input": "2a=3", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=3", "result": "a=\\frac{3}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{3}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=\\frac{3}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(\\frac{3}{2}\\right)^{2}\\:$$", "result": "-\\left(x^{2}+3x-1+\\left(\\frac{3}{2}\\right)^{2}-\\left(\\frac{3}{2}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}+3x+\\left(\\frac{3}{2}\\right)^{2}=\\left(x+\\frac{3}{2}\\right)^{2}$$", "Complete the square" ], "result": "-\\left(\\left(x+\\frac{3}{2}\\right)^{2}-1-\\left(\\frac{3}{2}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\left(x+\\frac{3}{2}\\right)^{2}+\\frac{13}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y>-\\left(x+\\frac{3}{2}\\right)^{2}+\\frac{13}{4}" }, { "type": "step", "primary": "Subtract $$\\frac{13}{4}$$ from both sides", "result": "y-\\frac{13}{4}>-\\left(x+\\frac{3}{2}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-1$$", "result": "-\\left(y-\\frac{13}{4}\\right)>\\left(x+\\frac{3}{2}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-\\frac{1}{4}\\right)\\left(y-\\frac{13}{4}\\right)=\\left(x-\\left(-\\frac{3}{2}\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-\\frac{1}{4}\\right)\\left(y-\\frac{13}{4}\\right)=\\left(x-\\left(-\\frac{3}{2}\\right)\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-\\frac{3}{2},\\:\\frac{13}{4}\\right),\\:p=-\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-\\frac{3}{2}))^{2}}{4(-\\frac{1}{4})}+\\frac{13}{4}", "displayFormula": "4(-\\frac{1}{4})(y-\\frac{13}{4})=(x-(-\\frac{3}{2}))^{2}", "attributes": { "color": "PURPLE", "lineType": "DASH", "isAsymptote": false } }, { "evalFormula": "y=\\frac{7}{2}", "displayFormula": "y=\\frac{7}{2}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-\\frac{3}{2},\\frac{13}{4})", "(-\\frac{3}{2},3)" ], "pointsDecimal": [ { "fst": -1.5, "snd": 3.25 }, { 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