y^2+4x+1=0

{ "query": { "display": "$$y^{2}+4x+1=0$$", "symbolab_question": "CONIC#y^{2}+4x+1=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-\\frac{1}{4},0),p=-1" }, "steps": { "type": "interim", "title": "$$y^{2}+4x+1=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-\\frac{1}{4},\\:0\\right),\\:$$and focal length $$|p|=1$$", "input": "y^{2}+4x+1=0", "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 $$y^{2}+4x+1=0\\:$$in the standard form:$${\\quad}4\\left(-1\\right)\\left(x-\\left(-\\frac{1}{4}\\right)\\right)=\\left(y-0\\right)^{2}$$", "input": "y^{2}+4x+1=0", "steps": [ { "type": "step", "primary": "Subtract $$y^{2}$$ from both sides", "result": "y^{2}+4x+1-y^{2}=0-y^{2}" }, { "type": "step", "primary": "Refine", "result": "4x+1=-y^{2}" }, { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{4x+1}{-1}=\\frac{-y^{2}}{-1}" }, { "type": "step", "primary": "Simplify", "result": "-4x-1=y^{2}" }, { "type": "step", "primary": "Factor $$-4$$", "result": "\\left(-4\\right)\\left(x+\\frac{-1}{-4}\\right)=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "\\left(-4\\right)\\left(x+\\frac{1}{4}\\right)=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-4}{4}\\left(x+\\frac{1}{4}\\right)=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-1\\right)\\left(x+\\frac{1}{4}\\right)=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-1\\right)\\left(x-\\left(-\\frac{1}{4}\\right)\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aRgRQFW0MqhVT4GbzWEc4smfnf+f7rxYzWg3st7WK0axzzcMHjSLmak70ycPvkgfLHlRs0F2lS8z71EJhfVfEQcoiLKrONfIA8+fB2d3wo49eajA7BpGfE9oODIxje87sUcw1ttPAkiUYG2MHwFsdafud2BBSI5iFqZd6pwGMxpN5Aod6Hr1Lp2e/29KhSgU7CCRPNHKioRGFeNUeDwBhP/ilKgkmpRap8RWThRhtKSB64Ml8b0Ki0WaGopRmMsmH0/Zy5mINpsQVQVsbT3LEg==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-\\frac{1}{4},\\:0\\right),\\:p=-1" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-1)(x-(-\\frac{1}{4}))}+0", "displayFormula": "4(-1)(x-(-\\frac{1}{4}))=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-1)(x-(-\\frac{1}{4}))}+0", "displayFormula": "4(-1)(x-(-\\frac{1}{4}))=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{3}{4}", "displayFormula": "x=\\frac{3}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-\\frac{1}{4},0)", "(-\\frac{5}{4},0)" ], "pointsDecimal": [ { "fst": -0.25, "snd": 0 }, { "fst": -1.25, "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(-1)(x-(-\\frac{1}{4}))=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -11.5, "xMax": 11, "yMin": -11.25, "yMax": 11.25 } }, "showViewLarger": true } } }