{
"query": {
"display": "$$\\left(x-3\\right)^{2}=8\\left(y-2\\right)$$",
"symbolab_question": "CONIC#(x-3)^{2}=8(y-2)"
},
"solution": {
"level": "PERFORMED",
"subject": "Geometry",
"topic": "Parabola",
"subTopic": "formula",
"default": "(h,k)=(3,2),p=2"
},
"steps": {
"type": "interim",
"title": "$$\\left(x-3\\right)^{2}=8\\left(y-2\\right):\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(3,\\:2\\right),\\:$$and focal length $$|p|=2$$",
"input": "\\left(x-3\\right)^{2}=8\\left(y-2\\right)",
"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 $$\\left(x-3\\right)^{2}=8\\left(y-2\\right)\\:$$in the standard form:$${\\quad}4\\cdot\\:2\\left(y-2\\right)=\\left(x-3\\right)^{2}$$",
"input": "\\left(x-3\\right)^{2}=8\\left(y-2\\right)",
"steps": [
{
"type": "step",
"primary": "Switch sides",
"result": "8\\left(y-2\\right)=\\left(x-3\\right)^{2}"
},
{
"type": "step",
"primary": "Factor $$8$$",
"result": "8\\left(y+\\frac{-16}{8}\\right)=\\left(x-3\\right)^{2}"
},
{
"type": "step",
"primary": "Simplify",
"result": "8\\left(y-2\\right)=\\left(x-3\\right)^{2}"
},
{
"type": "step",
"primary": "Factor $$4$$",
"result": "4\\cdot\\:\\frac{8}{4}\\left(y-2\\right)=\\left(x-3\\right)^{2}"
},
{
"type": "step",
"primary": "Simplify",
"result": "4\\cdot\\:2\\left(y-2\\right)=\\left(x-3\\right)^{2}"
},
{
"type": "step",
"primary": "Rewrite as",
"result": "4\\cdot\\:2\\left(y-2\\right)=\\left(x-3\\right)^{2}"
}
],
"meta": {
"interimType": "Parabola Canonical Format Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7x7c70iKgri0FohdBA3wVDeSV/4dmSIf2wz34h7AuPmOTdaV09PMxEKZ9FieghTFwIHjjrlQK96/oL7qEXVfedeEfpojVh0cP5vSMCDPkeM10BNumxmVU/tlY6yuoAdGPDQX0ZKJOCf04qYjohD4YtP8//6/nV5O4fb8Xgwi7maqIP2TH9VLmoizBQwdUXcYEkjY/D+BfdIj+0PqU+nm3Xwoz5I5Xp2hOBZ2vMDvjy2RpGwoPd870QszKiiHupZ++ihjOSo1xKKuRxYOAfpknTA=="
}
},
{
"type": "step",
"primary": "Therefore parabola properties are:",
"result": "\\left(h,\\:k\\right)=\\left(3,\\:2\\right),\\:p=2"
}
],
"meta": {
"solvingClass": "Parabola"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"funcsToDraw": {
"funcs": [
{
"evalFormula": "y=\\frac{(x-3)^{2}}{4\\cdot 2}+2",
"displayFormula": "4\\cdot 2(y-2)=(x-3)^{2}",
"attributes": {
"color": "PURPLE",
"lineType": "NORMAL",
"isAsymptote": false
}
},
{
"evalFormula": "y=0",
"displayFormula": "y=0",
"attributes": {
"color": "GRAY",
"lineType": "NORMAL",
"labels": [
"\\mathrm{directrix}"
],
"isAsymptote": false
}
}
]
},
"pointsToDraw": {
"pointsLatex": [
"(3,2)",
"(3,4)"
],
"pointsDecimal": [
{
"fst": 3,
"snd": 2
},
{
"fst": 3,
"snd": 4
}
],
"attributes": [
{
"color": "PURPLE",
"labels": [
"\\mathrm{vertex}"
],
"labelTypes": [
"DEFAULT"
],
"labelColors": [
"PURPLE"
]
},
{
"color": "PURPLE",
"labels": [
"\\mathrm{focus}"
],
"labelTypes": [
"DEFAULT"
],
"labelColors": [
"PURPLE"
]
}
]
},
"functionChanges": [
{
"origFormulaLatex": [],
"finalFormulaLatex": [],
"plotTitle": "4\\cdot 2(y-2)=(x-3)^{2}",
"paramsLatex": [],
"paramsReplacementsLatex": []
}
],
"localBoundingBox": {
"xMin": -19.5,
"xMax": 25.5,
"yMin": -20.5,
"yMax": 24.5
}
},
"showViewLarger": true
}
}
}
Solution
Solution
Solution steps
Rewrite in the standard form:
Therefore parabola properties are:
Graph
Popular Examples
Frequently Asked Questions (FAQ)
What is (x-3)^2=8(y-2) ?
The solution to (x-3)^2=8(y-2) is Parabola with (h,k)=(3,2),p=2