vertices f(x)=3x^2-18x+23

{ "query": { "display": "vertices $$f\\left(x\\right)=3x^{2}-18x+23$$", "symbolab_question": "CONIC#vertices f(x)=3x^{2}-18x+23" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(3,-4)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=3x^{2}-18x+23$$", "symbolab_question": "vertexquadratic y=3x^{2}-18x+23" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=3x^{2}-18x+23$$", "symbolab_question": "vertexparabola y=3x^{2}-18x+23" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=3x^{2}-18x+23$$", "symbolab_question": "vertexform y=3x^{2}-18x+23" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=3x^{2}-18x+23$$", "symbolab_question": "vertexzeros y=3x^{2}-18x+23" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=3x^{2}-18x+23:{\\quad}$$Minimum $$\\left(3,\\:-4\\right)$$", "input": "y=3x^{2}-18x+23", "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=23" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-18\\right)}{2\\cdot\\:3}" }, { "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}=3\\cdot\\:3^{2}-18\\cdot\\:3+23", "result": "y_{v}=-4", "steps": [ { "type": "interim", "title": "Simplify $$3\\cdot\\:3^{2}-18\\cdot\\:3+23:{\\quad}-4$$", "input": "3\\cdot\\:3^{2}-18\\cdot\\:3+23", "result": "y_{v}=-4", "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+23" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-54+23=-31$$", "result": "=3^{3}-31" }, { "type": "step", "primary": "$$3^{3}=27$$", "result": "=27-31" }, { "type": "step", "primary": "Subtract the numbers: $$27-31=-4$$", "result": "=-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70B7LWWq4sQas3ZX987qWKzxJ9Y3GJsxHV6ir9aomhxnehkKrn0era9rz8TlL+x/v/0We7dcrUw8Lb8omBGxhoO9sGZu5A1MXROmEpnxG69rWcTN4w49IeAiiPupUjF67vF79KycKnZffEdvblXjP6shKWDUXblDAfV4HnfyLGqU=" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(3,\\:-4\\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{Minimum}\\:\\left(3,\\:-4\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-3)^{2}}{4\\frac{1}{12}}-4", "displayFormula": "4\\frac{1}{12}(y-(-4))=(x-3)^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{49}{12}", "displayFormula": "y=-\\frac{49}{12}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(3,-4)", "(3,-\\frac{47}{12})" ], "pointsDecimal": [ { "fst": 3, "snd": -4 }, { "fst": 3, "snd": -3.9166666666666665 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { 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