-f(x)=4x^2+2x-2

{ "query": { "display": "$$-f\\left(x\\right)=4x^{2}+2x-2$$", "symbolab_question": "CONIC#-f(x)=4x^{2}+2x-2" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-\\frac{1}{4},\\frac{9}{4}),p=-\\frac{1}{16}" }, "steps": { "type": "interim", "title": "$$-y=4x^{2}+2x-2:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-\\frac{1}{4},\\:\\frac{9}{4}\\right),\\:$$and focal length $$|p|=\\frac{1}{16}$$", "input": "-y=4x^{2}+2x-2", "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 $$-y=4x^{2}+2x-2\\:$$in the standard form", "input": "-y=4x^{2}+2x-2", "steps": [ { "type": "step", "primary": "Divide by $$-1$$", "result": "y=-4x^{2}-2x+2" }, { "type": "interim", "title": "Complete the square $$-4x^{2}-2x+2:{\\quad}-4\\left(x+\\frac{1}{4}\\right)^{2}+\\frac{9}{4}$$", "input": "-4x^{2}-2x+2", "steps": [ { "type": "step", "primary": "Write $$-4x^{2}-2x+2\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-4$$" ], "result": "-4\\left(x^{2}+\\frac{x}{2}-\\frac{1}{2}\\right)" }, { "type": "interim", "title": "$$2a=\\frac{1}{2}{\\quad:\\quad}a=\\frac{1}{4}$$", "input": "2a=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=\\frac{1}{2}", "result": "a=\\frac{1}{4}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{\\frac{1}{2}}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2a}{2}=\\frac{\\frac{1}{2}}{2}", "result": "a=\\frac{1}{4}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2a}{2}:{\\quad}a$$", "input": "\\frac{2a}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=a" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7n/9TIosDfOyZXH4usT9jsy061ljBSPJeENOw2efoSWuFFDuzzgIzA1F36c+SAFGgRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pQyGeXuoJHAaVrX92ZjOTT" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{1}{2}}{2}:{\\quad}\\frac{1}{4}$$", "input": "\\frac{\\frac{1}{2}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{1}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{1}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajcW8I4vzSi5qkg7xcFXSuN0DnzlbPZjyKgy1eUCFsLd5xXDODtFBCC8Uf836IcE9x3E2Z5v/GFq2v9cvwjSfE0weKYFFhGodJZ4mS5Q+F9MLRBHnX1v95RIc5nuXPcUCV8q735ZaAWAgo4anwjJ6vBM=" } }, { "type": "step", "result": "a=\\frac{1}{4}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$\\left(\\frac{1}{4}\\right)^{2}\\:$$", "result": "-4\\left(x^{2}+\\frac{x}{2}-\\frac{1}{2}+\\left(\\frac{1}{4}\\right)^{2}-\\left(\\frac{1}{4}\\right)^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$x^{2}+\\frac{1}{2}x+\\left(\\frac{1}{4}\\right)^{2}=\\left(x+\\frac{1}{4}\\right)^{2}$$", "Complete the square" ], "result": "-4\\left(\\left(x+\\frac{1}{4}\\right)^{2}-\\frac{1}{2}-\\left(\\frac{1}{4}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-4\\left(x+\\frac{1}{4}\\right)^{2}+\\frac{9}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=-4\\left(x+\\frac{1}{4}\\right)^{2}+\\frac{9}{4}" }, { "type": "step", "primary": "Subtract $$\\frac{9}{4}$$ from both sides", "result": "y-\\frac{9}{4}=-4\\left(x+\\frac{1}{4}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-4$$", "result": "-\\frac{1}{4}\\left(y-\\frac{9}{4}\\right)=\\left(x+\\frac{1}{4}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-\\frac{1}{16}\\right)\\left(y-\\frac{9}{4}\\right)=\\left(x-\\left(-\\frac{1}{4}\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-\\frac{1}{16}\\right)\\left(y-\\frac{9}{4}\\right)=\\left(x-\\left(-\\frac{1}{4}\\right)\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-\\frac{1}{4},\\:\\frac{9}{4}\\right),\\:p=-\\frac{1}{16}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-\\frac{1}{4}))^{2}}{4(-\\frac{1}{16})}+\\frac{9}{4}", "displayFormula": 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