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

{ "query": { "display": "$$-f\\left(x\\right)=4x^{2}+4x-2$$", "symbolab_question": "CONIC#-f(x)=4x^{2}+4x-2" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-\\frac{1}{2},3),p=-\\frac{1}{16}" }, "steps": { "type": "interim", "title": "$$-y=4x^{2}+4x-2:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-\\frac{1}{2},\\:3\\right),\\:$$and focal length $$|p|=\\frac{1}{16}$$", "input": "-y=4x^{2}+4x-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 $$-y=4x^{2}+4x-2\\:$$in the standard form", "input": "-y=4x^{2}+4x-2", "steps": [ { "type": "step", "primary": "Divide by $$-1$$", "result": "y=-4x^{2}-4x+2" }, { "type": "interim", "title": "Complete the square $$-4x^{2}-4x+2:{\\quad}-4\\left(x+\\frac{1}{2}\\right)^{2}+3$$", "input": "-4x^{2}-4x+2", "steps": [ { "type": "step", "primary": "Write $$-4x^{2}-4x+2\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-4$$" ], "result": "-4\\left(x^{2}+x-\\frac{1}{2}\\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": "-4\\left(x^{2}+x-\\frac{1}{2}+\\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": "-4\\left(\\left(x+\\frac{1}{2}\\right)^{2}-\\frac{1}{2}-\\left(\\frac{1}{2}\\right)^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-4\\left(x+\\frac{1}{2}\\right)^{2}+3" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "y=-4\\left(x+\\frac{1}{2}\\right)^{2}+3" }, { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "y-3=-4\\left(x+\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-4$$", "result": "-\\frac{1}{4}\\left(y-3\\right)=\\left(x+\\frac{1}{2}\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-\\frac{1}{16}\\right)\\left(y-3\\right)=\\left(x-\\left(-\\frac{1}{2}\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-\\frac{1}{16}\\right)\\left(y-3\\right)=\\left(x-\\left(-\\frac{1}{2}\\right)\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-\\frac{1}{2},\\:3\\right),\\:p=-\\frac{1}{16}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-\\frac{1}{2}))^{2}}{4(-\\frac{1}{16})}+3", "displayFormula": "4(-\\frac{1}{16})(y-3)=(x-(-\\frac{1}{2}))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{49}{16}", "displayFormula": "y=\\frac{49}{16}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-\\frac{1}{2},3)", "(-\\frac{1}{2},\\frac{47}{16})" ], "pointsDecimal": [ { "fst": -0.5, "snd": 3 }, { "fst": -0.5, "snd": 2.9375 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-\\frac{1}{16})(y-3)=(x-(-\\frac{1}{2}))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -2.6428571428571432, "xMax": 2, "yMin": -0.9285714285714288, "yMax": 3.7142857142857144 } }, "showViewLarger": true } } }