intercepts of f(x)=-3x^2-x+4

{ "query": { "display": "intercepts $$f\\left(x\\right)=-3x^{2}-x+4$$", "symbolab_question": "CONIC#intercepts f(x)=-3x^{2}-x+4" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "intercepts", "default": "\\mathrm{X\\:Intercepts}: (-\\frac{4}{3},0),(1,0),\\mathrm{Y\\:Intercepts}: (0,4)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Axis interception points of $$-3x^{2}-x+4:\\quad\\:$$X Intercepts$$:\\:\\left(-\\frac{4}{3},\\:0\\right),\\:\\left(1,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:4\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$-3x^{2}-x+4:{\\quad}\\left(-\\frac{4}{3},\\:0\\right),\\:\\left(1,\\:0\\right)$$", "input": "-3x^{2}-x+4", "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 $$-3x^{2}-x+4=0:{\\quad}x=-\\frac{4}{3},\\:x=1$$", "input": "-3x^{2}-x+4=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "-3x^{2}-x+4=0", "result": "{x}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\left(-3\\right)\\cdot\\:4}}{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 $${\\quad}a=-3,\\:b=-1,\\:c=4$$", "result": "{x}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\left(-3\\right)\\cdot\\:4}}{2\\left(-3\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7eoy/W9yBXzH3G8ZC4aphvuAUJ8MrmDHxLcfnZ8t1JTOo/JI5bBgpgExN510TA5cywo6iOxOQeL70yTMBl67ndVrsdW4QlnVHZaYhaD1DKw8kGPbQ6vsS8SDpWj3qJP5uWKUPwrqBNjSXSd0f0a/FnI7XphmTRt6RoyOZ66xLGNYJbPb2kSAQiH8fEEM2S+s7zx97WZE62guOLkKRkZ4roXFxn1rssLY3m6zG42uRPCSQw7D74UJEEmspKsfbHPrGpmJ6ZAWKa2Xs9HhAOhjrjVe2nmAAeV1KiDzEE9dFq0Moc+zWfKHEYaTFWOEcCWrBSgSVf9gfAksz4HtFcy4+i+w/rOyd5aPv89U4pRBmcfn2/C/rdeFINnLLhSZZfcpl5FOU5DDFhEudchbE5CU+pX5QeOPnAIW5dOQ+zdaOF52FA8Qbs7VnVnRJjlwXtyy6VjzasxIZD+ag+oqDVOLPX4vXP5xxoSKrz0abbXUotpaSY/+cSinQeRR8DuBv3x5/JsReacQHo4DAXV3/DR5fpjLhLfVmzfJKOxhRZrQgYeBcd/7w88tI9ZQZ3vfejKK6a6dSLS2NRPKyKbA3Lglf0CS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-1\\right)^{2}-4\\left(-3\\right)\\cdot\\:4}=7$$", "input": "\\sqrt{\\left(-1\\right)^{2}-4\\left(-3\\right)\\cdot\\:4}", "result": "{x}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:7}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-1\\right)^{2}+4\\cdot\\:3\\cdot\\:4}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "interim", "title": "$$4\\cdot\\:3\\cdot\\:4=48$$", "input": "4\\cdot\\:3\\cdot\\:4", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3\\cdot\\:4=48$$", "result": "=48" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fVpMbrjjLKU8f2Xgsw8zZpf22UOvO0DOrnAFdEXB/LOjkVi15I8rBefLi4Iyt2wrCklV65q5uyJBBcrfcaVZqzATxmeBrt3pexYcISPTLogasATp3RyM8Uem+q1T0joU" } }, { "type": "step", "result": "=\\sqrt{1+48}" }, { "type": "step", "primary": "Add the numbers: $$1+48=49$$", "result": "=\\sqrt{49}" }, { "type": "step", "primary": "Factor the number: $$49=7^{2}$$", "result": "=\\sqrt{7^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{7^{2}}=7$$" ], "result": "=7", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gcCSkwHGkSzBJS28+axO64ZsLkmSavYl0gatjrrN7wAAlilG71elit3w1IBbYN0Pq2GRGjqZRXo7V9B2SJha6aN6Hv6MoTMtvtU0IQwXdn+SVpPUu8d2DohT7uf7kqbJPNR0fFfr6ezLntN0PrrG6yS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-\\left(-1\\right)+7}{2\\left(-3\\right)},\\:{x}_{2}=\\frac{-\\left(-1\\right)-7}{2\\left(-3\\right)}" }, { "type": "interim", "title": "$$x=\\frac{-\\left(-1\\right)+7}{2\\left(-3\\right)}:{\\quad}-\\frac{4}{3}$$", "input": "\\frac{-\\left(-1\\right)+7}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1+7}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Add the numbers: $$1+7=8$$", "result": "=\\frac{8}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{8}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{8}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{4}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cgJTBTBwsm6Rfg1t3BMzeOpMLhKW6Q7aKPmRj+2nNwJ1g99dC9fj9sg0EHzBIRDRICJwfdMN6lgbQtjUYnROKRSYeD6JDEaUbEpYNvzKYnPyfEhRNr+qR0uUiFep2cys3VO3ZCptS89+bCBOCRqFBA==" } }, { "type": "interim", "title": "$$x=\\frac{-\\left(-1\\right)-7}{2\\left(-3\\right)}:{\\quad}1$$", "input": "\\frac{-\\left(-1\\right)-7}{2\\left(-3\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1-7}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Subtract the numbers: $$1-7=-6$$", "result": "=\\frac{-6}{-2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{-6}{-6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{6}{6}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75hiL11gHjXKPxaUkyIy1R+pMLhKW6Q7aKPmRj+2nNwJ1g99dC9fj9sg0EHzBIRDR+8ZDu8iF4MSewt4yms1lIb3Tcan7wFkSuOUEIM4ZHqEKw73WmjgpRzlKcRkcJLlZJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=-\\frac{4}{3},\\:x=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(-\\frac{4}{3},\\:0\\right),\\:\\left(1,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7NoFnog1tKUNoxGjtAY+ueaeeFw+IDj27lwWcKVA3alsqWVX6xzKg+mqD2grc5O15qerEuL3gWPzyuK/0KZXp+wMdlpKImtKtq7R7fwRyOCVCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$-3x^{2}-x+4:{\\quad}\\left(0,\\:4\\right)$$", "input": "-3x^{2}-x+4", "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 $$y=-3\\cdot\\:0^{2}-0+4:{\\quad}y=4$$", "input": "y=-3\\cdot\\:0^{2}-0+4", "steps": [ { "type": "interim", "title": "Simplify $$-3\\cdot\\:0^{2}-0+4:{\\quad}4$$", "input": "-3\\cdot\\:0^{2}-0+4", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=-3\\cdot\\:0-0+4" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=-0-0+4" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-0-0+4=4$$", "result": "=4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gRyRfw+35bt189/6U5cNLoZC8stK7OxmrdCIKgX/DPrMwViaLUXkeD+JukROhWdjPyzWgCgEb2/nvTpNVPjY/h4pgUWEah0lniZLlD4X0wsxnG6ucCrqt2mfsJqEoHxjzsJy1RQqgiQlMe7ayex9mw==" } }, { "type": "step", "result": "y=4" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:4\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfnlowF6Rjnwbz+LySCtA1Oz2+x3mP6Ouka0XLQBn1LRUOlcBNNeXoGI0NE3bKqzubzsP6zsneWj7/PVOKUQZnH5BDxuQSYMIgSoM7v4yle7EQ==" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(-\\frac{4}{3},\\:0\\right),\\:\\left(1,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:4\\right)" } ], "meta": { "solvingClass": "Function Intersect" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-3x^{2}-x+4" }, "showViewLarger": true } }, "meta": { "showVerify": true } }