vertices y=18x^2-14x+9

{ "query": { "display": "vertices $$y=18x^{2}-14x+9$$", "symbolab_question": "CONIC#vertices y=18x^{2}-14x+9" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(\\frac{7}{18},\\frac{113}{18})", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=18x^{2}-14x+9$$", "symbolab_question": "vertexquadratic y=18x^{2}-14x+9" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=18x^{2}-14x+9$$", "symbolab_question": "vertexparabola y=18x^{2}-14x+9" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=18x^{2}-14x+9$$", "symbolab_question": "vertexform y=18x^{2}-14x+9" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=18x^{2}-14x+9:{\\quad}$$Minimum $$\\left(\\frac{7}{18},\\:\\frac{113}{18}\\right)$$", "input": "y=18x^{2}-14x+9", "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=18,\\:b=-14,\\:c=9" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-14\\right)}{2\\cdot\\:18}" }, { "type": "interim", "title": "Simplify $$-\\frac{-14}{2\\cdot\\:18}:{\\quad}\\frac{7}{18}$$", "input": "-\\frac{\\left(-14\\right)}{2\\cdot\\:18}", "result": "x_{v}=\\frac{7}{18}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:18=36$$", "result": "=-\\frac{-14}{36}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\left(-\\frac{14}{36}\\right)" }, { "type": "interim", "title": "Cancel $$\\frac{14}{36}:{\\quad}\\frac{7}{18}$$", "input": "\\frac{14}{36}", "result": "=-\\left(-\\frac{7}{18}\\right)", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{7}{18}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYk42rzUq7F/hFPuNEE4zqGd1g99dC9fj9sg0EHzBIRDRhunMHjbLs8kI6vtWE18ScVhWM3AOKGmUe/b+yXcRPxbV21gLEdaEWxRyo94byDTJZu/tsHp3BeUYauFMwHMf/L8yD3hLQ33B7/8/LpbPE3o=" } }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{7}{18}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fCQP9KhCM44QD3jHN6Xou+UKYxrDe1sGP6m+7Gf5JG0gJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ02l2APXXXDfYQ/2VRhRZQ0eNvb7k0sVmuwf19w9aD9NfgGlVmOKugLSTH2MFyXTR3klWNq8Gf2GuHCMJaNbu/OJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Plug in $$x_{v}=\\frac{7}{18}\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=18\\left(\\frac{7}{18}\\right)^{2}-14\\cdot\\:\\frac{7}{18}+9", "result": "y_{v}=\\frac{113}{18}", "steps": [ { "type": "interim", "title": "Simplify $$18\\left(\\frac{7}{18}\\right)^{2}-14\\cdot\\:\\frac{7}{18}+9:{\\quad}\\frac{113}{18}$$", "input": "18\\left(\\frac{7}{18}\\right)^{2}-14\\cdot\\:\\frac{7}{18}+9", "result": "y_{v}=\\frac{113}{18}", "steps": [ { "type": "interim", "title": "$$18\\left(\\frac{7}{18}\\right)^{2}=\\frac{49}{18}$$", "input": "18\\left(\\frac{7}{18}\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(\\frac{7}{18}\\right)^{2}=\\frac{7^{2}}{18^{2}}$$", "input": "\\left(\\frac{7}{18}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{7^{2}}{18^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7uFPHK/xUkK7mVQzBAXZ7g+iEPDD5lvIAC9CzFeUpV5JwkKGJWEPFPk38sdJMsyPIR3tJ1VOKVWJ040WYopBSoSL7Hvq8cp13yqWidBkjjBMbOU+G3xuL3mrjwpk6ibw6VLjotxnXOW3kXxLSNGNrpD0Tg4Kz6YlvAxXqDqazbFSwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=18\\cdot\\:\\frac{7^{2}}{18^{2}}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7^{2}\\cdot\\:18}{18^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$18$$", "result": "=\\frac{7^{2}}{18}" }, { "type": "step", "primary": "$$7^{2}=49$$", "result": "=\\frac{49}{18}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77Zo6bIiJRCLINMPTnU6TgqzSx8CEuHuQHCzxH53NrDEJQJZuTAY5js+oqjdT8kslQGl4psGJToEIYMRtqg/5IatsDHCGc0gnJHdHN33VLWkBFNxrVEk7IOYn4MW1vuftDHTQBzMV3IZY+cU+iX4oFDdxtxE4Lq29/mB4lDGC4GI=" } }, { "type": "interim", "title": "$$14\\cdot\\:\\frac{7}{18}=\\frac{49}{9}$$", "input": "14\\cdot\\:\\frac{7}{18}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{7\\cdot\\:14}{18}" }, { "type": "step", "primary": "Multiply the numbers: $$7\\cdot\\:14=98$$", "result": "=\\frac{98}{18}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{49}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wYJ9xXupfR1Avje3BteH61XS97rz0CWCFGPBW3EA+p51g99dC9fj9sg0EHzBIRDRQIWE1LCrU4RkDbItPPlXsdbFeyVnN0lOd0ddEzgC8f6PkB5uUyPkQTFtvwztDsTX3iQWsadlGnVUxTZ8zWn6klfMNKWXtgO6oMegxHqtOfE=" } }, { "type": "step", "result": "=\\frac{49}{18}-\\frac{49}{9}+9" }, { "type": "step", "primary": "Convert element to fraction: $$9=\\frac{9}{1}$$", "result": "=\\frac{9}{1}+\\frac{49}{18}-\\frac{49}{9}" }, { "type": "interim", "title": "Least Common Multiplier of $$1,\\:18,\\:9:{\\quad}18$$", "input": "1,\\:18,\\:9", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "step", "primary": "Prime factorization of $$1$$", "meta": { "solvingClass": "Composite Integer" } }, { "type": "interim", "title": "Prime factorization of $$18:{\\quad}2\\cdot\\:3\\cdot\\:3$$", "input": "18", "steps": [ { "type": "step", "primary": "$$18\\:$$divides by $$2\\quad\\:18=9\\cdot\\:2$$", "result": "=2\\cdot\\:9" }, { "type": "step", "primary": "$$9\\:$$divides by $$3\\quad\\:9=3\\cdot\\:3$$", "result": "=2\\cdot\\:3\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:3\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdyuEeNBgSa3LrIvx33A/jwUB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIECMyNL+vYlkZN7gSrf1sNmp" } }, { "type": "interim", "title": "Prime factorization of $$9:{\\quad}3\\cdot\\:3$$", "input": "9", "steps": [ { "type": "step", "primary": "$$9\\:$$divides by $$3\\quad\\:9=3\\cdot\\:3$$", "result": "=3\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRjcq2F6aCU5dakzncJqkgGc/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp02ygGGsmIGE3wDrLHDVisx" } }, { "type": "step", "primary": "Compute a number comprised of factors that appear in at least one of the following:<br/>$$1,\\:18,\\:9$$", "result": "=2\\cdot\\:3\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3\\cdot\\:3=18$$", "result": "=18" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$18$$" }, { "type": "step", "primary": "For $$\\frac{9}{1}:\\:$$multiply the denominator and numerator by $$18$$", "result": "\\frac{9}{1}=\\frac{9\\cdot\\:18}{1\\cdot\\:18}=\\frac{162}{18}" }, { "type": "step", "primary": "For $$\\frac{49}{9}:\\:$$multiply the denominator and numerator by $$2$$", "result": "\\frac{49}{9}=\\frac{49\\cdot\\:2}{9\\cdot\\:2}=\\frac{98}{18}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{162}{18}+\\frac{49}{18}-\\frac{98}{18}" }, { "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{162+49-98}{18}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$162+49-98=113$$", "result": "=\\frac{113}{18}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77Zo6bIiJRCLINMPTnU6Tgi1DZhuW7tHQbw6xpBiuIWEFUgyYjIZ/Cyhc0b6dJHJMAJYpRu9XpYrd8NSAW2DdD/KxLrO04AooUAReaJjhZCbnDdFCpmgkc8hvMlsQatc872wZm7kDUxdE6YSmfEbr2nVRc8yFkYN/irVBM1Pk3g8KK5J3jjpvrQVGl6twypWzMzVePhNz/csQv+GQTxBP/CGa9Nnh0SUYHMHdoJH0Nl8=" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(\\frac{7}{18},\\:\\frac{113}{18}\\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/>$$a=18$$", "result": "\\mathrm{Minimum}\\:\\left(\\frac{7}{18},\\:\\frac{113}{18}\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { 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