vertices f(x)=-3x^2+18x-8

{ "query": { "display": "vertices $$f\\left(x\\right)=-3x^{2}+18x-8$$", "symbolab_question": "CONIC#vertices f(x)=-3x^{2}+18x-8" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Maximum}\\:(3,19)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=-3x^{2}+18x-8$$", "symbolab_question": "vertexquadratic y=-3x^{2}+18x-8" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=-3x^{2}+18x-8$$", "symbolab_question": "vertexparabola y=-3x^{2}+18x-8" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=-3x^{2}+18x-8$$", "symbolab_question": "vertexform y=-3x^{2}+18x-8" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=-3x^{2}+18x-8$$", "symbolab_question": "vertexzeros y=-3x^{2}+18x-8" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=-3x^{2}+18x-8:{\\quad}$$Maximum $$\\left(3,\\:19\\right)$$", "input": "y=-3x^{2}+18x-8", "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=-3,\\:b=18,\\:c=-8" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{18}{2\\left(-3\\right)}" }, { "type": "interim", "title": "Simplify $$-\\frac{18}{2\\left(-3\\right)}:{\\quad}3$$", "input": "-\\frac{18}{2\\left(-3\\right)}", "result": "x_{v}=3", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{18}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=-\\frac{18}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\left(-\\frac{18}{6}\\right)" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{18}{6}=3$$", "result": "=-\\left(-3\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RtxjuGHamb8yp1ORZOVDebM1yKzQWXXYFS09QAKLmt7MwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jx4pgUWEah0lniZLlD4X0wtSf3pTTy+YYQVrKzFN7mUna2okK4HEtoi0OpViBcfAiQ==" } }, { "type": "interim", "title": "Plug in $$x_{v}=3\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=-3\\cdot\\:3^{2}+18\\cdot\\:3-8", "result": "y_{v}=19", "steps": [ { "type": "interim", "title": "Simplify $$-3\\cdot\\:3^{2}+18\\cdot\\:3-8:{\\quad}19$$", "input": "-3\\cdot\\:3^{2}+18\\cdot\\:3-8", "result": "y_{v}=19", "steps": [ { "type": "interim", "title": "$$3\\cdot\\:3^{2}=3^{3}$$", "input": "3\\cdot\\:3^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$3\\cdot\\:3^{2}=\\:3^{1+2}$$" ], "result": "=3^{1+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=3^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Hms94GO2lPhP86CRLnF5ZS061ljBSPJeENOw2efoSWsQTQNYsb/dJGjOWnJrejA1/z//r+dXk7h9vxeDCLuZqr/a44LhFnPi3hnaJemt/8jII+mBznc+g6xGnMwA8n4l" } }, { "type": "interim", "title": "$$18\\cdot\\:3=54$$", "input": "18\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$18\\cdot\\:3=54$$", "result": "=54" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75iXBtX2/Il5hpgUl9hnoiyAn9lkDfZkicUGkO3EF+IqCgDgxxcrQEo5KrPMPckaifvJ0JoMFys4yEED0JNmYai50wz63FpzuEtXxIb6OKc8=" } }, { "type": "step", "result": "=-3^{3}+54-8" }, { "type": "step", "primary": "Add/Subtract the numbers: $$54-8=46$$", "result": "=46-3^{3}" }, { "type": "step", "primary": "$$3^{3}=27$$", "result": "=46-27" }, { "type": "step", "primary": "Subtract the numbers: $$46-27=19$$", "result": "=19" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vOdj406Ua4DRB8rJNsqrZBJFxs0K/E9HR37PaTfY75nehkKrn0era9rz8TlL+x/vVifJ8VdKs2k/AIlbmIu58+9sGZu5A1MXROmEpnxG69pU9CqZ7lZvYEOqUIk+j4vz/ohcON6gZWxWx1UnBonhHQcZVM+afZNRv3WMpbQMT+s=" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(3,\\:19\\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/>$$a=-3$$", "result": "\\mathrm{Maximum}\\:\\left(3,\\:19\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-3)^{2}}{4(-\\frac{1}{12})}+19", "displayFormula": "4(-\\frac{1}{12})(y-19)=(x-3)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{229}{12}", "displayFormula": "y=\\frac{229}{12}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(3,19)", "(3,\\frac{227}{12})" ], "pointsDecimal": [ { "fst": 3, "snd": 19 }, { "fst": 3, "snd": 18.916666666666668 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { 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