vertices y=4x^2-40x+98

{ "query": { "display": "vertices $$y=4x^{2}-40x+98$$", "symbolab_question": "CONIC#vertices y=4x^{2}-40x+98" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(5,-2)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=4x^{2}-40x+98$$", "symbolab_question": "vertexquadratic y=4x^{2}-40x+98" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=4x^{2}-40x+98$$", "symbolab_question": "vertexparabola y=4x^{2}-40x+98" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=4x^{2}-40x+98$$", "symbolab_question": "vertexform y=4x^{2}-40x+98" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=4x^{2}-40x+98$$", "symbolab_question": "vertexzeros y=4x^{2}-40x+98" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=4x^{2}-40x+98:{\\quad}$$Minimum $$\\left(5,\\:-2\\right)$$", "input": "y=4x^{2}-40x+98", "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=4,\\:b=-40,\\:c=98" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-40\\right)}{2\\cdot\\:4}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=5" }, { "type": "interim", "title": "Plug in $$x_{v}=5\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=4\\cdot\\:5^{2}-40\\cdot\\:5+98", "result": "y_{v}=-2", "steps": [ { "type": "interim", "title": "Simplify $$4\\cdot\\:5^{2}-40\\cdot\\:5+98:{\\quad}-2$$", "input": "4\\cdot\\:5^{2}-40\\cdot\\:5+98", "result": "y_{v}=-2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$40\\cdot\\:5=200$$", "result": "=5^{2}\\cdot\\:4-200+98" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-200+98=-102$$", "result": "=5^{2}\\cdot\\:4-102" }, { "type": "interim", "title": "$$5^{2}\\cdot\\:4=100$$", "input": "5^{2}\\cdot\\:4", "steps": [ { "type": "step", "primary": "$$5^{2}=25$$", "result": "=25\\cdot\\:4" }, { "type": "step", "primary": "Multiply the numbers: $$25\\cdot\\:4=100$$", "result": "=100" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X34HpdvyeskLuHl/N/aV9i061ljBSPJeENOw2efoSWu65fCi1HoNHTJyfeKngzdhYumNlYD1wFrfO7PZFchQjo5ilINgrE4JvEDLW9k6FvmwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=100-102" }, { "type": "step", "primary": "Subtract the numbers: $$100-102=-2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7beFEjAMf4+irg0UR7wKM8Y7GhpGkxWzPVuwvlXL+MwLehkKrn0era9rz8TlL+x/vTt+XLrGGFP41O9wAUvwUq+9sGZu5A1MXROmEpnxG69qt4movs/wPf6iZqsxc5tOdSHHSGssWudHrQgJc8bhsRZbMznkcHMtVU8yflx/jSA8=" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(5,\\:-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/>$$a=4$$", "result": "\\mathrm{Minimum}\\:\\left(5,\\:-2\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-5)^{2}}{4\\frac{1}{16}}-2", "displayFormula": "4\\frac{1}{16}(y-(-2))=(x-5)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{33}{16}", "displayFormula": "y=-\\frac{33}{16}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(5,-2)", "(5,-\\frac{31}{16})" ], "pointsDecimal": [ { "fst": 5, "snd": -2 }, { "fst": 5, "snd": -1.9375 } ], "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}{16}(y-(-2))=(x-5)^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -1.5, "xMax": 6, "yMin": -4.714285714285714, "yMax": 2.7857142857142856 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }