vertices y=(x-1)^2-2

{ "query": { "display": "vertices $$y=\\left(x-1\\right)^{2}-2$$", "symbolab_question": "CONIC#vertices y=(x-1)^{2}-2" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(1,-2)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=\\left(x-1\\right)^{2}-2$$", "symbolab_question": "vertexquadratic y=(x-1)^{2}-2" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=\\left(x-1\\right)^{2}-2$$", "symbolab_question": "vertexparabola y=(x-1)^{2}-2" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=\\left(x-1\\right)^{2}-2$$", "symbolab_question": "vertexform y=(x-1)^{2}-2" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=\\left(x-1\\right)^{2}-2$$", "symbolab_question": "vertexzeros y=(x-1)^{2}-2" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=\\left(x-1\\right)^{2}-2:{\\quad}$$Minimum $$\\left(1,\\:-2\\right)$$", "input": "y=\\left(x-1\\right)^{2}-2", "steps": [ { "type": "definition", "title": "Parabola equation in polynomial form", "text": "The vertex of an up-down facing parabola of the form $$y=ax^2+bx+c\\:$$is $$x_{v}=-\\frac{b}{2a}$$" }, { "type": "interim", "title": "Rewrite $$y=\\left(x-1\\right)^{2}-2\\:$$in the form $$y=ax^{2}+bx+c$$", "input": "y=\\left(x-1\\right)^{2}-2", "result": "y=x^{2}-2x-1", "steps": [ { "type": "interim", "title": "Expand $$\\left(x-1\\right)^{2}-2:{\\quad}x^{2}-2x-1$$", "input": "\\left(x-1\\right)^{2}-2", "result": "y=x^{2}-2x-1", "steps": [ { "type": "interim", "title": "$$\\left(x-1\\right)^{2}:{\\quad}x^{2}-2x+1$$", "result": "=x^{2}-2x+1-2", "steps": [ { "type": "step", "primary": "Apply Perfect Square Formula: $$\\left(a-b\\right)^{2}=a^{2}-2ab+b^{2}$$", "secondary": [ "$$a=x,\\:\\:b=1$$" ], "meta": { "practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square", "practiceTopic": "Expand Perfect Square" } }, { "type": "step", "result": "=x^{2}-2x\\cdot\\:1+1^{2}" }, { "type": "interim", "title": "Simplify $$x^{2}-2x\\cdot\\:1+1^{2}:{\\quad}x^{2}-2x+1$$", "input": "x^{2}-2x\\cdot\\:1+1^{2}", "result": "=x^{2}-2x+1", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=x^{2}-2\\cdot\\:1\\cdot\\:x+1" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=x^{2}-2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Add/Subtract the numbers: $$1-2=-1$$", "result": "=x^{2}-2x-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Expand Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nPUcGeyVwG+WLq+E0NhbeNuY5182ZeybmovaunlLabHeRv0Qg6eY1kN4TiDkMoxs+yxfSSpJ0xijNZCv3R7nSdbA+zX4bD3u3gx65o2NJhM++SOLO3jH8jrj67jNwAqx6OqIDAyjCGquPrqgZgDIDQ==" } } ], "meta": { "interimType": "Rewrite In Form 2Eq" } }, { "type": "step", "primary": "The parabola parameters are:", "result": "a=1,\\:b=-2,\\:c=-1" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-2\\right)}{2\\cdot\\:1}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=1" }, { "type": "interim", "title": "Plug in $$x_{v}=1\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=1^{2}-2\\cdot\\:1-1", "result": "y_{v}=-2", "steps": [ { "type": "step", "primary": "Simplify", "result": "y_{v}=-2" } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(1,\\:-2\\right)" }, { "type": "step", "primary": "If $$a<0,\\:$$then the vertex is a maximum value<br/>If $$a>0,\\:$$then the vertex is a minimum value<br/>$$a=1$$", "result": "\\mathrm{Minimum}\\:\\left(1,\\:-2\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-1)^{2}}{4\\frac{1}{4}}-2", "displayFormula": "4\\frac{1}{4}(y-(-2))=(x-1)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{9}{4}", "displayFormula": "y=-\\frac{9}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,-2)", "(1,-\\frac{7}{4})" ], "pointsDecimal": [ { "fst": 1, "snd": -2 }, { "fst": 1, "snd": -1.75 } ], "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 \\frac{1}{4}(y-(-2))=(x-1)^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -1.9696428571428575, "xMax": 3.6553571428571425, "yMin": -4.098214285714286, "yMax": 1.5267857142857144 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }