y=-3x^2-12x+7

{ "query": { "display": "$$y=-3x^{2}-12x+7$$", "symbolab_question": "FUNCTION#y=-3x^{2}-12x+7" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "Combination", "default": "\\mathrm{Domain}: -\\infty <x<\\infty <br/>\\mathrm{Range}: f(x)\\le 19<br/>\\mathrm{X\\:Intercepts}: (-\\frac{6+\\sqrt{57}}{3},0),(\\frac{\\sqrt{57}-6}{3},0),\\mathrm{Y\\:Intercepts}: (0,7)<br/>\\mathrm{Vertex}: \\mathrm{Maximum}\\:(-2,19)<br/>\\mathrm{Inverse}: -\\frac{12+\\sqrt{-12x+228}}{6},-\\frac{12-\\sqrt{-12x+228}}{6}", "interval": "\\mathrm{Domain}: (-\\infty ,\\infty )<br/>\\mathrm{Range}: (-\\infty ,19]" }, "steps": { "type": "interim", "steps": [ { "type": "interim", "title": "Domain of $$-3x^{2}-12x+7\\::{\\quad}-\\infty\\:<x<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<x<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Range of $$-3x^{2}-12x+7:{\\quad}f\\left(x\\right)\\le\\:19$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "interim", "title": "Vertex of $$-3x^{2}-12x+7:{\\quad}$$Maximum $$\\left(-2,\\:19\\right)$$", "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=-12,\\:c=7" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-12\\right)}{2\\left(-3\\right)}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-2" }, { "type": "interim", "title": "Plug in $$x_{v}=-2\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=-3\\left(-2\\right)^{2}-12\\left(-2\\right)+7", "result": "y_{v}=19", "steps": [ { "type": "interim", "title": "Simplify $$-3\\left(-2\\right)^{2}-12\\left(-2\\right)+7:{\\quad}19$$", "input": "-3\\left(-2\\right)^{2}-12\\left(-2\\right)+7", "result": "y_{v}=19", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=-3\\left(-2\\right)^{2}+12\\cdot\\:2+7" }, { "type": "interim", "title": "$$\\left(-2\\right)^{2}=2^{2}$$", "input": "\\left(-2\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sNuhSfBo+/I8oqMceBlhcs0ag8T1MwTer44+aCS/ZFBDeoKWfP4f0hW8hp+DjlqkWG48kfKlXwh1JXHkPaftrOeZImDuB9kLWbJJECF6RjY=" } }, { "type": "step", "result": "=-2^{2}\\cdot\\:3+12\\cdot\\:2+7" }, { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:2=24$$", "result": "=-2^{2}\\cdot\\:3+24+7" }, { "type": "step", "primary": "Add the numbers: $$24+7=31$$", "result": "=31-2^{2}\\cdot\\:3" }, { "type": "interim", "title": "$$2^{2}\\cdot\\:3=12$$", "input": "2^{2}\\cdot\\:3", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YlMCdSu9defMDx6stnf0hS061ljBSPJeENOw2efoSWvRhv/4tiXq5Z5AYo1OPkf//COFlUvA93NcQfHx1F5YKhX7Z8MQIFQnyR+DcLAqjH0kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=31-12" }, { "type": "step", "primary": "Subtract the numbers: $$31-12=19$$", "result": "=19" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D7USg0FwTt/JqcO5sf4inkwBUTcstXEhNu1GvAHxhDRwkKGJWEPFPk38sdJMsyPI0y4JUu6wBfb1inK49IQCqpjcBIL5pmo83UMFZRSzJsXxWYYsDJXMALmOcQ49SORjZ+hB8FZZQtfFNOOZqs3JTg==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-2,\\: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(-2,\\:19\\right)" } ], "meta": { "solvingClass": "Function Vertex", "interimType": "Range Parabola Find Vertex 1Eq" } }, { "type": "step", "primary": "For a parabola $$ax^2+bx+c\\:$$with Vertex $$\\left(x_v,\\:y_v\\right)$$<br/>$$\\quad$$If $$a<0\\:$$the range is $$f\\left(x\\right)\\le\\:y_v$$<br/>$$\\quad$$If $$a>0\\:$$the range is $$f\\left(x\\right)\\ge\\:y_v$$<br/>$$a=-3,\\:$$Vertex $$\\left(x_v,\\:y_v\\right)=\\left(-2,\\:19\\right)$$", "result": "f\\left(x\\right)\\le\\:19" } ], "meta": { "solvingClass": "Function Range", "interimType": "Function Range Top 1Eq" } }, { "type": "interim", "title": "Axis interception points of $$-3x^{2}-12x+7:\\quad\\:$$X Intercepts$$:\\:\\left(-\\frac{6+\\sqrt{57}}{3},\\:0\\right),\\:\\left(\\frac{\\sqrt{57}-6}{3},\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:7\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$-3x^{2}-12x+7:{\\quad}\\left(-\\frac{6+\\sqrt{57}}{3},\\:0\\right),\\:\\left(\\frac{\\sqrt{57}-6}{3},\\:0\\right)$$", "input": "-3x^{2}-12x+7", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$-3x^{2}-12x+7=0:{\\quad}x=-\\frac{6+\\sqrt{57}}{3},\\:x=\\frac{\\sqrt{57}-6}{3}$$", "input": "-3x^{2}-12x+7=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-3x^{2}-12x+7=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\cdot\\:7}}{2\\left(-3\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-3,\\:b=-12,\\:c=7$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\cdot\\:7}}{2\\left(-3\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\cdot\\:7}=2\\sqrt{57}$$", "input": "\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\cdot\\:7}", "result": "{x}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:2\\sqrt{57}}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-12\\right)^{2}+4\\cdot\\:3\\cdot\\:7}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-12\\right)^{2}=12^{2}$$" ], "result": "=\\sqrt{12^{2}+4\\cdot\\:3\\cdot\\:7}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3\\cdot\\:7=84$$", "result": "=\\sqrt{12^{2}+84}" }, { "type": "step", "primary": "$$12^{2}=144$$", "result": "=\\sqrt{144+84}" }, { "type": "step", "primary": "Add the numbers: $$144+84=228$$", "result": "=\\sqrt{228}" }, { "type": "interim", "title": "Prime factorization of $$228:{\\quad}2^{2}\\cdot\\:3\\cdot\\:19$$", "input": "228", "result": "=\\sqrt{2^{2}\\cdot\\:3\\cdot\\:19}", "steps": [ { "type": "step", "primary": "$$228\\:$$divides by $$2\\quad\\:228=114\\cdot\\:2$$", "result": "=2\\cdot\\:114" }, { "type": "step", "primary": "$$114\\:$$divides by $$2\\quad\\:114=57\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:57" }, { "type": "step", "primary": "$$57\\:$$divides by $$3\\quad\\:57=19\\cdot\\:3$$", "result": "=2\\cdot\\:2\\cdot\\:3\\cdot\\:19" }, { "type": "step", "primary": "$$2,\\:3,\\:19$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:3\\cdot\\:19" }, { "type": "step", "result": "=2^{2}\\cdot\\:3\\cdot\\:19" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFz1ZOvWlO8xhH3978xYNl1l9NqQ+lATTQCpcI1REFC/BKGFkX1ACFJO8XX3CmdEmA1Hl6Av5y4W8WjpVXaDxbN/" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2^{2}}\\sqrt{3\\cdot\\:19}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{3\\cdot\\:19}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{57}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fr9oMXl8QtE4/ZTpPJTfWwYi3HfucZnTOs72TF4PpGV8kR7hsO/rTOTBE0w4+r1R1Jetp9AvoQcY2QSDwFJhz3nrBahx7psmPAqF8EZlVn9y5sKdhk4nkvz/aCx3KbDUNAQ0fs5d4cIzG9IojYDToXJuRhr1suF+R24KBd7OxHz09b8ScebSC3w3U4GdspHO" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-\\left(-12\\right)+2\\sqrt{57}}{2\\left(-3\\right)},\\:{x}_{2}=\\frac{-\\left(-12\\right)-2\\sqrt{57}}{2\\left(-3\\right)}" }, { "type": "interim", "title": "$$x=\\frac{-\\left(-12\\right)+2\\sqrt{57}}{2\\left(-3\\right)}:{\\quad}-\\frac{6+\\sqrt{57}}{3}$$", "input": "\\frac{-\\left(-12\\right)+2\\sqrt{57}}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{12+2\\sqrt{57}}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{12+2\\sqrt{57}}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{12+2\\sqrt{57}}{6}" }, { "type": "interim", "title": "Cancel $$\\frac{12+2\\sqrt{57}}{6}:{\\quad}\\frac{6+\\sqrt{57}}{3}$$", "input": "\\frac{12+2\\sqrt{57}}{6}", "result": "=-\\frac{6+\\sqrt{57}}{3}", "steps": [ { "type": "interim", "title": "Factor $$12+2\\sqrt{57}:{\\quad}2\\left(6+\\sqrt{57}\\right)$$", "input": "12+2\\sqrt{57}", "result": "=\\frac{2\\left(6+\\sqrt{57}\\right)}{6}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:6+2\\sqrt{57}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(6+\\sqrt{57}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{6+\\sqrt{57}}{3}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmSpVSfSBN2uA/EHg4CTRe7bSuPgUruiTXFi0sbOTR1azvYaGmwZlvy5qMc82VI6dTafmUlB40DvEI5tbobmseXIH1t1C0KVEu72nhBJsKDKtL6zr3uPMOsrthH7FUU3zs/LQkknSI+9n5+R6K9RM00zpgOZO7OU/8U9s9IbANqnvzIPeEtDfcHv/z8uls8Teg==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7anucnun7OX2G4yim0upPTtBnz7XNr7j7lh2iGSwkZF1OzT4XyDhGkKKDfKL1SdPro5FYteSPKwXny4uCMrdsK7Y6vmDqKwZf+0fcbYi+t7TNdpbwuS5+BaBzxuKiurudvkRs4+YUhUNkIqvm7zANekqNIiH2fjf/k/GM559dQ2h9fgwRToS2w8uUn0v+5P3r" } }, { "type": "interim", "title": "$$x=\\frac{-\\left(-12\\right)-2\\sqrt{57}}{2\\left(-3\\right)}:{\\quad}\\frac{\\sqrt{57}-6}{3}$$", "input": "\\frac{-\\left(-12\\right)-2\\sqrt{57}}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{12-2\\sqrt{57}}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{12-2\\sqrt{57}}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "secondary": [ "$$12-2\\sqrt{57}=-\\left(2\\sqrt{57}-12\\right)$$" ], "result": "=\\frac{2\\sqrt{57}-12}{6}" }, { "type": "interim", "title": "Factor $$2\\sqrt{57}-12:{\\quad}2\\left(\\sqrt{57}-6\\right)$$", "input": "2\\sqrt{57}-12", "result": "=\\frac{2\\left(\\sqrt{57}-6\\right)}{6}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\sqrt{57}-2\\cdot\\:6" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(\\sqrt{57}-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{\\sqrt{57}-6}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7E4vVNnhePDnnZC3zSqgvwtBnz7XNr7j7lh2iGSwkZF1OzT4XyDhGkKKDfKL1SdPro5FYteSPKwXny4uCMrdsK7aAFzbG1f/QRHFPkdb7LEb71HGbwPrRhlOG+b3IfKv38AoJ8UcsRHaJEUK5RS7pU5QZN6Ux1kNxJHPnLwRNFGdqGKuGeZ9XW6VCfM+H6JA1" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=-\\frac{6+\\sqrt{57}}{3},\\:x=\\frac{\\sqrt{57}-6}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(-\\frac{6+\\sqrt{57}}{3},\\:0\\right),\\:\\left(\\frac{\\sqrt{57}-6}{3},\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7MGywl9E0mTKA0uD/nEU1dEmLHHwEVdW6UyKBXafhy+UlpYZc2BwfzhOOX4RpvgNLN28Bz4sms00qkQRBmsujP5XVZgJOkz9/sKpzBl+qLFsssIKV6us6bHwxKGlcjfNsI=" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$-3x^{2}-12x+7:{\\quad}\\left(0,\\:7\\right)$$", "input": "-3x^{2}-12x+7", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=-3\\cdot\\:0^{2}-12\\cdot\\:0+7:{\\quad}y=7$$", "input": "y=-3\\cdot\\:0^{2}-12\\cdot\\:0+7", "steps": [ { "type": "interim", "title": "Simplify $$-3\\cdot\\:0^{2}-12\\cdot\\:0+7:{\\quad}7$$", "input": "-3\\cdot\\:0^{2}-12\\cdot\\:0+7", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=-3\\cdot\\:0-12\\cdot\\:0+7" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=-0-0+7" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-0-0+7=7$$", "result": "=7" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gRyRfw+35bt189/6U5cNLpenMg2UHki//DKc+l9BJNHehkKrn0era9rz8TlL+x/vTE4OONfGJ7J2yqTcD4S3boEFMST8lDZxn1Yq5HMKVTvCjP6Gpd21LYp5Rjoi+XdWEG5SWzOsH3NUmPSsIHLxZCVpCjTkTqqI0SNA8bJoHUU=" } }, { "type": "step", "result": "y=7" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:7\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfkiral2LRBDCIuqBNRSfwJAed90Mu+pdqaYRvPrfemtfTCJtI9vvSu8YNGFp1F5HQLHLGAZaUd20nBvmobuFTth3bVcUjwpcy4+piKewfJp+rCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(-\\frac{6+\\sqrt{57}}{3},\\:0\\right),\\:\\left(\\frac{\\sqrt{57}-6}{3},\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:7\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Function Intercepts Top 2Eq" } }, { "type": "interim", "title": "Vertex of $$-3x^{2}-12x+7:{\\quad}$$Maximum $$\\left(-2,\\:19\\right)$$", "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=-12,\\:c=7" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-12\\right)}{2\\left(-3\\right)}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-2" }, { "type": "interim", "title": "Plug in $$x_{v}=-2\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=-3\\left(-2\\right)^{2}-12\\left(-2\\right)+7", "result": "y_{v}=19", "steps": [ { "type": "interim", "title": "Simplify $$-3\\left(-2\\right)^{2}-12\\left(-2\\right)+7:{\\quad}19$$", "input": "-3\\left(-2\\right)^{2}-12\\left(-2\\right)+7", "result": "y_{v}=19", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=-3\\left(-2\\right)^{2}+12\\cdot\\:2+7" }, { "type": "interim", "title": "$$\\left(-2\\right)^{2}=2^{2}$$", "input": "\\left(-2\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sNuhSfBo+/I8oqMceBlhcs0ag8T1MwTer44+aCS/ZFBDeoKWfP4f0hW8hp+DjlqkWG48kfKlXwh1JXHkPaftrOeZImDuB9kLWbJJECF6RjY=" } }, { "type": "step", "result": "=-2^{2}\\cdot\\:3+12\\cdot\\:2+7" }, { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:2=24$$", "result": "=-2^{2}\\cdot\\:3+24+7" }, { "type": "step", "primary": "Add the numbers: $$24+7=31$$", "result": "=31-2^{2}\\cdot\\:3" }, { "type": "interim", "title": "$$2^{2}\\cdot\\:3=12$$", "input": "2^{2}\\cdot\\:3", "steps": [ { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4\\cdot\\:3" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=12" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YlMCdSu9defMDx6stnf0hS061ljBSPJeENOw2efoSWvRhv/4tiXq5Z5AYo1OPkf//COFlUvA93NcQfHx1F5YKhX7Z8MQIFQnyR+DcLAqjH0kt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=31-12" }, { "type": "step", "primary": "Subtract the numbers: $$31-12=19$$", "result": "=19" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D7USg0FwTt/JqcO5sf4inkwBUTcstXEhNu1GvAHxhDRwkKGJWEPFPk38sdJMsyPI0y4JUu6wBfb1inK49IQCqpjcBIL5pmo83UMFZRSzJsXxWYYsDJXMALmOcQ49SORjZ+hB8FZZQtfFNOOZqs3JTg==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-2,\\: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(-2,\\:19\\right)" } ], "meta": { "solvingClass": "Function Vertex", "interimType": "Range Parabola Find Vertex 1Eq" } }, { "type": "interim", "title": "Inverse of $$-3x^{2}-12x+7:{\\quad}-\\frac{12+\\sqrt{-12x+228}}{6},\\:-\\frac{12-\\sqrt{-12x+228}}{6}$$", "steps": [ { "type": "definition", "title": "Function Inverse definition", "text": "A function g is the inverse of function f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$" }, { "type": "step", "result": "y=-3x^{2}-12x+7" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=-3x^{2}-12x+7", "result": "x=-3y^{2}-12y+7", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=-3y^{2}-12y+7" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZZc+1yXDR7kO6DlrdmsqarP1QEHiDWNC4JIhSQp4rMahJhmHmOn1ZYO09JecLYwWnjqrDy3NDe4NVvKyniJVdeTtaIGhePqqK3BBTAo4+xqnUtKPDaodmxiuo7IrXzNElgMJyP5jWUfdIrCWzTQsJoE=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=-3y^{2}-12y+7$$", "input": "x=-3y^{2}-12y+7", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-3y^{2}-12y+7=x" }, { "type": "interim", "title": "Move $$x\\:$$to the left side", "input": "-3y^{2}-12y+7=x", "result": "-3y^{2}-12y+7-x=0", "steps": [ { "type": "step", "primary": "Subtract $$x$$ from both sides", "result": "-3y^{2}-12y+7-x=x-x" }, { "type": "step", "primary": "Simplify", "result": "-3y^{2}-12y+7-x=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-3y^{2}-12y+7-x=0", "result": "{y}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\left(7-x\\right)}}{2\\left(-3\\right)}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=-3,\\:b=-12,\\:c=7-x$$", "result": "{y}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\left(7-x\\right)}}{2\\left(-3\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "Simplify $$\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\left(7-x\\right)}:{\\quad}\\sqrt{-12x+228}$$", "input": "\\sqrt{\\left(-12\\right)^{2}-4\\left(-3\\right)\\left(7-x\\right)}", "result": "{y}_{1,\\:2}=\\frac{-\\left(-12\\right)\\pm\\:\\sqrt{-12x+228}}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-12\\right)^{2}+4\\cdot\\:3\\left(7-x\\right)}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-12\\right)^{2}=12^{2}$$" ], "result": "=\\sqrt{12^{2}+4\\cdot\\:3\\left(-x+7\\right)}" }, { "type": "step", "primary": "Refine", "result": "=\\sqrt{144+12\\left(-x+7\\right)}" }, { "type": "interim", "title": "Expand $$144+12\\left(7-x\\right):{\\quad}-12x+228$$", "input": "144+12\\left(7-x\\right)", "result": "=\\sqrt{-12x+228}", "steps": [ { "type": "interim", "title": "Expand $$12\\left(7-x\\right):{\\quad}84-12x$$", "input": "12\\left(7-x\\right)", "result": "=144+84-12x", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=12,\\:b=7,\\:c=x$$" ], "result": "=12\\cdot\\:7-12x", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$12\\cdot\\:7=84$$", "result": "=84-12x" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GhnZO29lcjiMLaVYY1dC2gOfOVs9mPIqDLV5QIWwt3mP7Im2S12pfb4Np3PdaE4b72wZm7kDUxdE6YSmfEbr2g1xhrnDTmI2mCoonDTpyxzABdlXv7JDi2LIYmKGSTz1" } }, { "type": "step", "primary": "Add the numbers: $$144+84=228$$", "result": "=-12x+228" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AeidSeR6Ndv01UsK3nE0Nt6GQqufR6tr2vPxOUv7H+9vPkSo5r329bUiRaBlCGDw9WTr1pTvMYR9/e/MWDZdZYJy8vaNLPDVWVFwZfiRrqq+aTHrZUGydc2vpXZFPPypJLd1ohke2Wgml78++2zI0g==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Separate the solutions", "result": "{y}_{1}=\\frac{-\\left(-12\\right)+\\sqrt{-12x+228}}{2\\left(-3\\right)},\\:{y}_{2}=\\frac{-\\left(-12\\right)-\\sqrt{-12x+228}}{2\\left(-3\\right)}" }, { "type": "interim", "title": "$$y=\\frac{-\\left(-12\\right)+\\sqrt{-12x+228}}{2\\left(-3\\right)}:{\\quad}-\\frac{12+\\sqrt{-12x+228}}{6}$$", "input": "\\frac{-\\left(-12\\right)+\\sqrt{-12x+228}}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{12+\\sqrt{-12x+228}}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{12+\\sqrt{-12x+228}}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{12+\\sqrt{-12x+228}}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7anucnun7OX2G4yim0upPToaSE+Yk8Kvh5GU0b6Ex4bjqTC4SlukO2ij5kY/tpzcCdYPfXQvX4/bINBB8wSEQ0daa6BHuy+JYokwH+JgZUeqGkhPmJPCr4eRlNG+hMeG40O8Gm1BsGKvsbBn4aVXxL69bfF8VrKipME/R2pSOO0ddawmZoWoTuAB8rTKl9A0QdW0CoiKhw1EpTv6Ngx6HPA==" } }, { "type": "interim", "title": "$$y=\\frac{-\\left(-12\\right)-\\sqrt{-12x+228}}{2\\left(-3\\right)}:{\\quad}-\\frac{12-\\sqrt{-12x+228}}{6}$$", "input": "\\frac{-\\left(-12\\right)-\\sqrt{-12x+228}}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{12-\\sqrt{-12x+228}}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{12-\\sqrt{-12x+228}}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{12-\\sqrt{-12x+228}}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7E4vVNnhePDnnZC3zSqgvwoaSE+Yk8Kvh5GU0b6Ex4bjqTC4SlukO2ij5kY/tpzcCdYPfXQvX4/bINBB8wSEQ0eLaZEp969KuHTTxT8BwWEuGkhPmJPCr4eRlNG+hMeG40O8Gm1BsGKvsbBn4aVXxL69bfF8VrKipME/R2pSOO0eGGGLkWkG6Y6EhqRXCi7cPdW0CoiKhw1EpTv6Ngx6HPA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "y=-\\frac{12+\\sqrt{-12x+228}}{6},\\:y=-\\frac{12-\\sqrt{-12x+228}}{6}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "-\\frac{12+\\sqrt{-12x+228}}{6},\\:-\\frac{12-\\sqrt{-12x+228}}{6}" } ], "meta": { "solvingClass": "Function Inverse", "interimType": "Function Inverse Top 1Eq" } } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-3x^{2}-12x+7" }, "showViewLarger": true } } }