vertices f(x)=x^2-2x-15

{ "query": { "display": "vertices $$f\\left(x\\right)=x^{2}-2x-15$$", "symbolab_question": "CONIC#vertices f(x)=x^{2}-2x-15" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(1,-16)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=x^{2}-2x-15$$", "symbolab_question": "vertexquadratic y=x^{2}-2x-15" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=x^{2}-2x-15$$", "symbolab_question": "vertexparabola y=x^{2}-2x-15" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=x^{2}-2x-15$$", "symbolab_question": "vertexform y=x^{2}-2x-15" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=x^{2}-2x-15$$", "symbolab_question": "vertexzeros y=x^{2}-2x-15" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=x^{2}-2x-15:{\\quad}$$Minimum $$\\left(1,\\:-16\\right)$$", "input": "y=x^{2}-2x-15", "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": "step", "primary": "The parabola parameters are:", "result": "a=1,\\:b=-2,\\:c=-15" }, { "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\\:1-15", "result": "y_{v}=-16", "steps": [ { "type": "step", "primary": "Simplify", "result": "y_{v}=-16" } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(1,\\:-16\\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,\\:-16\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-1)^{2}}{4\\frac{1}{4}}-16", "displayFormula": "4\\frac{1}{4}(y-(-16))=(x-1)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{65}{4}", "displayFormula": "y=-\\frac{65}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,-16)", "(1,-\\frac{63}{4})" ], "pointsDecimal": [ { "fst": 1, "snd": -16 }, { "fst": 1, "snd": -15.75 } ], "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-(-16))=(x-1)^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -11.299999999999999, "xMax": 12.985714285714286, "yMin": -19.42857142857143, "yMax": 4.857142857142856 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }