-9x-3y^2=6

{ "query": { "display": "$$-9x-3y^{2}=6$$", "symbolab_question": "CONIC#-9x-3y^{2}=6" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-\\frac{2}{3},0),p=-\\frac{3}{4}" }, "steps": { "type": "interim", "title": "$$-9x-3y^{2}=6:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-\\frac{2}{3},\\:0\\right),\\:$$and focal length $$|p|=\\frac{3}{4}$$", "input": "-9x-3y^{2}=6", "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 $$-9x-3y^{2}=6\\:$$in the standard form:$${\\quad}4\\left(-\\frac{3}{4}\\right)\\left(x-\\left(-\\frac{2}{3}\\right)\\right)=\\left(y-0\\right)^{2}$$", "input": "-9x-3y^{2}=6", "steps": [ { "type": "step", "primary": "Add $$3y^{2}$$ to both sides", "result": "-9x-3y^{2}+3y^{2}=6+3y^{2}" }, { "type": "step", "primary": "Refine", "result": "-9x=6+3y^{2}" }, { "type": "step", "primary": "Subtract $$6$$ from both sides", "result": "-9x-6=6+3y^{2}-6" }, { "type": "step", "primary": "Refine", "result": "-9x-6=3y^{2}" }, { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{-9x-6}{3}=\\frac{3y^{2}}{3}" }, { "type": "step", "primary": "Simplify", "result": "-3x-2=y^{2}" }, { "type": "step", "primary": "Factor $$-3$$", "result": "\\left(-3\\right)\\left(x+\\frac{-2}{-3}\\right)=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "\\left(-3\\right)\\left(x+\\frac{2}{3}\\right)=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-3}{4}\\left(x+\\frac{2}{3}\\right)=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-\\frac{3}{4}\\right)\\left(x+\\frac{2}{3}\\right)=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-\\frac{3}{4}\\right)\\left(x-\\left(-\\frac{2}{3}\\right)\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yTxDh034SWhZ5ueVIbVjw5CSbGFPWhWMuZIMKkBJ4lyxzzcMHjSLmak70ycPvkgfLHlRs0F2lS8z71EJhfVfEQcoiLKrONfIA8+fB2d3wo6Z5DbXsf4Yp7miHKLOt1dFzH3dXoqGCinFfA7QshfpoDqwIYntN54hgQG/ljjxuqURztKE552b3ssyeALQtWRjdllRtXcbobcHL3OfgLNk6O3sW8S9ygXbUTm3dWG2zyNNTGhbJadPPNBGBjLoQfbEVVjP9e0t4R9KhD4d9/fAzL8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-\\frac{2}{3},\\:0\\right),\\:p=-\\frac{3}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-\\frac{3}{4})(x-(-\\frac{2}{3}))}+0", "displayFormula": "4(-\\frac{3}{4})(x-(-\\frac{2}{3}))=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-\\frac{3}{4})(x-(-\\frac{2}{3}))}+0", "displayFormula": "4(-\\frac{3}{4})(x-(-\\frac{2}{3}))=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{1}{12}", "displayFormula": "x=\\frac{1}{12}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-\\frac{2}{3},0)", "(-\\frac{17}{12},0)" ], "pointsDecimal": [ { "fst": -0.6666666666666666, "snd": 0 }, { "fst": -1.4166666666666667, "snd": 0 } ], "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{3}{4})(x-(-\\frac{2}{3}))=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -9.104166666666666, "xMax": 7.770833333333334, "yMin": -8.4375, "yMax": 8.4375 } }, "showViewLarger": true } } }