directrix x^2=-4y

{ "query": { "display": "directrix $$x^{2}=-4y$$", "symbolab_question": "CONIC#directrix x^{2}=-4y" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "directrix", "default": "y=1" }, "steps": { "type": "interim", "title": "Parabola directrix given $$x^{2}=-4y:{\\quad}y=1$$", "steps": [ { "type": "definition", "title": "Parabola Directrix", "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": "$$4p\\left(y-k\\right)=\\left(x-h\\right)^{2}\\:$$ 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 $$x^{2}=-4y\\:$$in the standard form:$${\\quad}4\\left(-1\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}$$", "input": "x^{2}=-4y", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-4y=x^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-4}{4}y=x^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-1\\right)y=x^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-1\\right)\\left(y-0\\right)=\\left(x-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J936ILcLgU5g+ZJLg3C3NONeTXjeP8Fry4AS6VRPW/JxnSd9ckJvhaVSw11yD/KmDHfDBptXBowSMDY4Q9A3bJC7kY3S3HkkoV2HG4XOf8zxR8atLdKAEWBYTg7tj/cnYb10NmRcKzn4LGDVg28k09SC0GmxzxcSnDJuZfVuuvR82YU/y4erwCxl2ICWmc/FCxb/e5ztW12ZFofTcvHh0O6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:p=-1" }, { "type": "step", "primary": "Parabola is symmetric around the y-axis and so the directrix is a line parallel to the x-axis, a distance $$-p$$ from the center $$\\left(0,\\:0\\right)$$ y-coordinate ", "result": "y=0-p" }, { "type": "step", "result": "y=0-\\left(-1\\right)" }, { "type": "step", "primary": "Refine", "result": "y=1" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x^{2}}{4(-1)}+0", "displayFormula": "4(-1)y=x^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=1", "displayFormula": "y=1", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,0)", "(0,-1)" ], "pointsDecimal": [ { "fst": 0, "snd": 0 }, { "fst": 0, "snd": -1 } ], "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)(y)=x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -11.25, "xMax": 11.25, "yMin": -11.25, "yMax": 11.25 } }, "showViewLarger": true } } }