4<=-x^2-y

{ "query": { "display": "$$4\\le\\:-x^{2}-y$$", "symbolab_question": "CONIC#4\\le -x^{2}-y" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(0,-4),p=-\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$4\\le\\:-x^{2}-y:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(0,\\:-4\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "4\\le\\:-x^{2}-y", "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 $$4\\le\\:-x^{2}-y\\:$$in the standard form:$${\\quad}4\\left(-\\frac{1}{4}\\right)\\left(y-\\left(-4\\right)\\right)=\\left(x-0\\right)^{2}$$", "input": "4\\le\\:-x^{2}-y", "steps": [ { "type": "step", "primary": "Add $$y$$ to both sides", "result": "4+y\\le\\:-x^{2}-y+y" }, { "type": "step", "primary": "Refine", "result": "4+y\\le\\:-x^{2}" }, { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{4+y}{-1}\\le\\:\\frac{-x^{2}}{-1}" }, { "type": "step", "primary": "Simplify", "result": "-4-y\\ge\\:x^{2}" }, { "type": "step", "primary": "Factor $$-1$$", "result": "\\left(-1\\right)\\left(y+\\frac{-4}{-1}\\right)\\ge\\:x^{2}" }, { "type": "step", "primary": "Simplify", "result": "\\left(-1\\right)\\left(y+4\\right)\\ge\\:x^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{-1}{4}\\left(y+4\\right)\\ge\\:x^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\left(-\\frac{1}{4}\\right)\\left(y+4\\right)\\ge\\:x^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\left(-\\frac{1}{4}\\right)\\left(y-\\left(-4\\right)\\right)=\\left(x-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pwKD6ZbvXGMjF+zPZcK4z5ib/XvrUvBWqCaaSCF6BQAvAnqnrwGiugLGrHSv5m+bshnS/Lmb44dkvLfuNuNJlm0RlOY2FmsyIIlSotqy8yRWMb8giXGtiwi3zKcDhCMgFhWH+ShBUUBxbtdiSIC+i9j8MQKdAE9pQr/EIpAoJoTWwPs1+Gw97t4MeuaNjSYTdF+uDWMMJvJ+uLEBcX8jgNLqfWb3DoQBaXH0uuAfyxb1EO8ZHk97q5ibJA6WFBDEzV8wKnOX1NcR15NGrcNYFA==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:-4\\right),\\:p=-\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{x^{2}}{4(-\\frac{1}{4})}-4", "displayFormula": "4(-\\frac{1}{4})(y-(-4))=x^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{15}{4}", "displayFormula": "y=-\\frac{15}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(0,-4)", "(0,-\\frac{17}{4})" ], "pointsDecimal": [ { "fst": 0, "snd": -4 }, { "fst": 0, "snd": -4.25 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "fills": [ { "ranges": [], "funcIndices": [], "funcs": [], "xIneq": false, "yIneq": false, "twoVar": true, "trueAboveLine": true, "color": "rgba(171, 181, 235, 0.3)", "func": "y=\\frac{x^{2}}{4(-\\frac{1}{4})}-4" } ], "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4(-\\frac{1}{4})(y-(-4))\\ge x^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -3.5714285714285716, "xMax": 3.5714285714285716, "yMin": -5.714285714285714, "yMax": 1.4285714285714288 } }, "showViewLarger": true } } }