vertices 3x^2+2x+1

{ "query": { "display": "vertices $$3x^{2}+2x+1$$", "symbolab_question": "CONIC#vertices 3x^{2}+2x+1" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(-\\frac{1}{3},\\frac{2}{3})", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=3x^{2}+2x+1$$", "symbolab_question": "vertexquadratic y=3x^{2}+2x+1" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=3x^{2}+2x+1$$", "symbolab_question": "vertexparabola y=3x^{2}+2x+1" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=3x^{2}+2x+1$$", "symbolab_question": "vertexform y=3x^{2}+2x+1" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=3x^{2}+2x+1:{\\quad}$$Minimum $$\\left(-\\frac{1}{3},\\:\\frac{2}{3}\\right)$$", "input": "y=3x^{2}+2x+1", "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=2,\\:c=1" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{2}{2\\cdot\\:3}" }, { "type": "interim", "title": "Simplify $$-\\frac{2}{2\\cdot\\:3}:{\\quad}-\\frac{1}{3}$$", "input": "-\\frac{2}{2\\cdot\\:3}", "result": "x_{v}=-\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=-\\frac{2}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Qd/qmI8YqGcT3DcQvOrjgETA1/h8RxUF2BM5nk5gvioJQJZuTAY5js+oqjdT8kslBU+VtQsVB1irMDE5QVw3Lv8//6/nV5O4fb8Xgwi7mapyhd7tjiG+GxQNxDvGkZUlQVIGjT+fO0q33tUlUvjj9wLmUBzjmbrTqEeDdMcVy4M=" } }, { "type": "interim", "title": "Plug in $$x_{v}=-\\frac{1}{3}\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=3\\left(-\\frac{1}{3}\\right)^{2}+2\\left(-\\frac{1}{3}\\right)+1", "result": "y_{v}=\\frac{2}{3}", "steps": [ { "type": "interim", "title": "Simplify $$3\\left(-\\frac{1}{3}\\right)^{2}+2\\left(-\\frac{1}{3}\\right)+1:{\\quad}\\frac{2}{3}$$", "input": "3\\left(-\\frac{1}{3}\\right)^{2}+2\\left(-\\frac{1}{3}\\right)+1", "result": "y_{v}=\\frac{2}{3}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=3\\left(-\\frac{1}{3}\\right)^{2}-2\\cdot\\:\\frac{1}{3}+1" }, { "type": "interim", "title": "$$3\\left(-\\frac{1}{3}\\right)^{2}=\\frac{1}{3}$$", "input": "3\\left(-\\frac{1}{3}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(-\\frac{1}{3}\\right)^{2}=\\frac{1}{3^{2}}$$", "input": "\\left(-\\frac{1}{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-\\frac{1}{3}\\right)^{2}=\\left(\\frac{1}{3}\\right)^{2}$$" ], "result": "=\\left(\\frac{1}{3}\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{1^{2}}{3^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\frac{1}{3^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rawoVK5DYfduDQOtaQNmR+iEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPIzFYCX4N1yHBrO2v8Z+i1OwH2kDe5DGYTz3TrPquGdIiSCzj6N4cFWSu9B9qajQ1ym+VCGiSwJq1rcjIpOBa/vPWOx8SqMXI3+MSIEx5BQA0=" } }, { "type": "step", "result": "=3\\cdot\\:\\frac{1}{3^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3}{3^{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=\\frac{3}{3^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D/BGy1bWboSwO4M7q3mxzUBgnzAZkUy+vMtgHHTjQTCrju+5Z51e/ZZSD3gRHwjB0Udyok/apayF+Hy+qOOanWYGynyltByXt5a4b+QbEK0iyDhkgHzCaiFGkvEV06VLJWH8eJYdxy7IEX51NUcPoLCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{3}=\\frac{2}{3}$$", "input": "2\\cdot\\:\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{3}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvijFIXl/yolDRhu0cz3GobRqrju+5Z51e/ZZSD3gRHwjB0/HYRNx78bNoDYGVrAhHUt7IItnasRPF1orTdtJLofZ4l49CKdXaF0TyCQkBTrtkkA08abkr2aJDMiCJis7DULCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{1}{3}-\\frac{2}{3}+1" }, { "type": "interim", "title": "Combine the fractions $$\\frac{1}{3}-\\frac{2}{3}:{\\quad}-\\frac{1}{3}$$", "result": "=-\\frac{1}{3}+1", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1-2}{3}" }, { "type": "step", "primary": "Subtract the numbers: $$1-2=-1$$", "result": "=\\frac{-1}{3}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{3}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Convert element to fraction: $$1=\\frac{1\\cdot\\:3}{3}$$", "result": "=\\frac{1\\cdot\\:3}{3}-\\frac{1}{3}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:3-1}{3}" }, { "type": "interim", "title": "$$1\\cdot\\:3-1=2$$", "input": "1\\cdot\\:3-1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:3=3$$", "result": "=3-1" }, { "type": "step", "primary": "Subtract the numbers: $$3-1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NgJleVMs0UCBhOVqEk7Ycd6GQqufR6tr2vPxOUv7H+9hLZhL3VORCH2mYsYpGjlNP/n/sT8Hudl/0KJRqY9qee021l+wa5+se0sDbP/2KgY=" } }, { "type": "step", "result": "=\\frac{2}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D/BGy1bWboSwO4M7q3mxzUvGIsTFh/OJc6yvI2aBGy/mzHVlU2rAjSf0VUOaH95KA585Wz2Y8ioMtXlAhbC3efMXbR2RE4IDn3zdYG7zid0UmHg+iQxGlGxKWDb8ymJzHimBRYRqHSWeJkuUPhfTCwQgBTCJEGzvprPB0upy4NytsglcecNQSZ3XxyJpr+uRAbROZ2EWOmlhD4bFxzWnuQ==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-\\frac{1}{3},\\:\\frac{2}{3}\\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(-\\frac{1}{3},\\:\\frac{2}{3}\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-\\frac{1}{3}))^{2}}{4\\frac{1}{12}}+\\frac{2}{3}", "displayFormula": "4\\frac{1}{12}(y-\\frac{2}{3})=(x-(-\\frac{1}{3}))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=\\frac{7}{12}", "displayFormula": "y=\\frac{7}{12}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-\\frac{1}{3},\\frac{2}{3})", "(-\\frac{1}{3},\\frac{3}{4})" ], "pointsDecimal": [ { "fst": -0.3333333333333333, "snd": 0.6666666666666666 }, { "fst": -0.3333333333333333, "snd": 0.75 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { 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