{
"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": "$$4\\le\\:-x^{2}-y:\\quad$$Parábola con vértice en $$\\left(h,\\:k\\right)=\\left(0,\\:-4\\right),\\:$$y longitud focal $$|p|=\\frac{1}{4}$$",
"input": "4\\le\\:-x^{2}-y",
"steps": [
{
"type": "definition",
"title": "Ecuación general de la parábola",
"text": "$$4p\\left(y-k\\right)=\\left(x-h\\right)^{2}\\:$$ es la ecuación general de la parábola cuando esta se abre hacia arriba, con vértice en $$\\left(h,\\:k\\right),\\:$$<br/>y longitud focal $$|p|$$"
},
{
"type": "interim",
"title": "Reescribir $$4\\le\\:-x^{2}-y\\:$$con la forma de la ecuación general de la parábola:$${\\quad}4\\left(-\\frac{1}{4}\\right)\\left(y-\\left(-4\\right)\\right)=\\left(x-0\\right)^{2}$$",
"input": "4\\le\\:-x^{2}-y",
"steps": [
{
"type": "step",
"primary": "Sumar $$y$$ a ambos lados",
"result": "4+y\\le\\:-x^{2}-y+y"
},
{
"type": "step",
"primary": "Simplificar",
"result": "4+y\\le\\:-x^{2}"
},
{
"type": "step",
"primary": "Dividir ambos lados entre $$-1$$",
"result": "\\frac{4+y}{-1}\\le\\:\\frac{-x^{2}}{-1}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "-4-y\\ge\\:x^{2}"
},
{
"type": "step",
"primary": "Factorizar $$-1$$",
"result": "\\left(-1\\right)\\left(y+\\frac{-4}{-1}\\right)\\ge\\:x^{2}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "\\left(-1\\right)\\left(y+4\\right)\\ge\\:x^{2}"
},
{
"type": "step",
"primary": "Factorizar $$4$$",
"result": "4\\cdot\\:\\frac{-1}{4}\\left(y+4\\right)\\ge\\:x^{2}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "4\\left(-\\frac{1}{4}\\right)\\left(y+4\\right)\\ge\\:x^{2}"
},
{
"type": "step",
"primary": "Reescribir como",
"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+Gw97t4MeuaNjSYT0us/p+JzoMbLPFxuc10WcwfjbsZIgEKkxWlpe5kanjYoCqrqNAj2G2/I/7i06ce7OXmXhRadLzepIKvQGYJML4hhfSfR2EteNLoJSzTlb9VfbhTv9fsqkPQHw1pUMaKa"
}
},
{
"type": "step",
"primary": "Por lo tanto, las propiedades de la parábola son:",
"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
}
}
}
Solución
Solución
Pasos de solución
Reescribir con la forma de la ecuación general de la parábola:
Por lo tanto, las propiedades de la parábola son: