vertices f(x)=x^2+6x

{ "query": { "display": "vertices $$f\\left(x\\right)=x^{2}+6x$$", "symbolab_question": "CONIC#vertices f(x)=x^{2}+6x" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(-3,-9)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=x^{2}+6x$$", "symbolab_question": "vertexquadratic y=x^{2}+6x" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=x^{2}+6x$$", "symbolab_question": "vertexparabola y=x^{2}+6x" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=x^{2}+6x$$", "symbolab_question": "vertexform y=x^{2}+6x" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=x^{2}+6x$$", "symbolab_question": "vertexzeros y=x^{2}+6x" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=x^{2}+6x:{\\quad}$$Minimum $$\\left(-3,\\:-9\\right)$$", "input": "y=x^{2}+6x", "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=6,\\:c=0" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{6}{2\\cdot\\:1}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-3" }, { "type": "interim", "title": "Plug in $$x_{v}=-3\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=\\left(-3\\right)^{2}+6\\left(-3\\right)", "result": "y_{v}=-9", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-3\\right)^{2}+6\\left(-3\\right):{\\quad}-9$$", "input": "\\left(-3\\right)^{2}+6\\left(-3\\right)", "result": "y_{v}=-9", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\left(-3\\right)^{2}-6\\cdot\\:3" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-3\\right)^{2}=3^{2}$$" ], "result": "=3^{2}-6\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:3=18$$", "result": "=3^{2}-18" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=9-18" }, { "type": "step", "primary": "Subtract the numbers: $$9-18=-9$$", "result": "=-9" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AlACI9PQGrF2WCxMIydHXACWKUbvV6WK3fDUgFtg3Q/6LssHhowubKmDqlSsH5z+ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz1RDuuXj/tUKW6U5X6Rxb4WsIjaxJ4DvjTb2fbKjbvtlQ==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-3,\\:-9\\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(-3,\\:-9\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-3))^{2}}{4\\frac{1}{4}}-9", "displayFormula": "4\\frac{1}{4}(y-(-9))=(x-(-3))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{37}{4}", "displayFormula": "y=-\\frac{37}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-3,-9)", "(-3,-\\frac{35}{4})" ], "pointsDecimal": [ { "fst": -3, "snd": -9 }, { "fst": -3, "snd": -8.75 } ], "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-(-9))=(x-(-3))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -8.857142857142858, "xMax": 5.428571428571428, "yMin": -11.428571428571429, "yMax": 2.8571428571428577 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }