foci y^2=-2x

{ "query": { "display": "foci $$y^{2}=-2x$$", "symbolab_question": "CONIC#foci y^{2}=-2x" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "foci", "default": "(-\\frac{1}{2},0)" }, "steps": { "type": "interim", "title": "Parabola focus given $$y^{2}=-2x:{\\quad}\\left(-\\frac{1}{2},\\:0\\right)$$", "steps": [ { "type": "definition", "title": "Parabola Focus", "text": "A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (the directrix)" }, { "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 $$y^{2}=-2x\\:$$in the standard form:$${\\quad}4\\left(-\\frac{1}{2}\\right)\\left(x-0\\right)=\\left(y-0\\right)^{2}$$", "input": "y^{2}=-2x", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-2x=y^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-2}{4}x=y^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-\\frac{1}{2}\\right)x=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-\\frac{1}{2}\\right)\\left(x-0\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aRgRQFW0MqhVT4GbzWEc4gPzfcrQwqpbdVOmllMsY7xeTXjeP8Fry4AS6VRPW/JxnSd9ckJvhaVSw11yD/KmDHfDBptXBowSMDY4Q9A3bJDasI5bAgvkB99F0ziQtvhJuX7fuMbxlCZKvOrjOW2rP/8//6/nV5O4fb8Xgwi7maqIP2TH9VLmoizBQwdUXcYE5AhrizZ3hCZ7G0NtzrdyrQMDmMFWkZD2KZqjQuE/LxJMyblZyz31NMRaqlLJV7CY" } }, { "type": "step", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-\\frac{1}{2}" }, { "type": "step", "primary": "Parabola is symmetric around the x-axis and so the focus lies a distance $$p$$ from the center $$\\left(0,\\:0\\right)$$ along the x-axis ", "result": "\\left(0+p,\\:0\\right)" }, { "type": "step", "result": "=\\left(0+\\left(-\\frac{1}{2}\\right),\\:0\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(-\\frac{1}{2},\\:0\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-\\frac{1}{2})x}+0", "displayFormula": "4(-\\frac{1}{2})x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-\\frac{1}{2})x}+0", "displayFormula": "4(-\\frac{1}{2})x=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{1}{2}", "displayFormula": "x=\\frac{1}{2}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(-\\frac{1}{2},0)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": -0.5, "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{1}{2})(x)=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -5.625, "xMax": 5.625, "yMin": -5.625, "yMax": 5.625 } }, "showViewLarger": true } } }