asymptotes of f(x)=(x^2-9)/(3x+6)

{ "query": { "display": "asymptotes $$f\\left(x\\right)=\\frac{x^{2}-9}{3x+6}$$", "symbolab_question": "FUNCTION#asymptotes f(x)=\\frac{x^{2}-9}{3x+6}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=-2,\\mathrm{Slant}: y=\\frac{1}{3}x-\\frac{2}{3}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{x^{2}-9}{3x+6}:\\quad\\:$$Vertical$$:\\:x=-2,\\:$$Slant$$:\\:y=\\frac{1}{3}x-\\frac{2}{3}$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{x^{2}-9}{3x+6}:{\\quad}x=-2$$", "input": "\\frac{x^{2}-9}{3x+6}", "steps": [ { "type": "definition", "title": "Vertical asymptotes of rational Functions", "text": "For rational functions, the vertical asymptotes are the undefined points, also known as the zeros of the denominator, of the simplified function." }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}x=-2$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{x^{2}-9}{3x+6}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$3x+6=0:{\\quad}x=-2$$", "input": "3x+6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "3x+6=0", "result": "3x=-6", "steps": [ { "type": "step", "primary": "Subtract $$6$$ from both sides", "result": "3x+6-6=0-6" }, { "type": "step", "primary": "Simplify", "result": "3x=-6" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Cx4OWLWWJoue/xH3nV87zhyOUtErWNRXbTqqjj69E3ugl/0UF6jVkzQeum9OSrU8RKvwpCsMm/dIDWKdnSUJtL1yD3gfmCfeqiwfHLReClq8VmH5b+IMwMCjiIIju0VR3CmpoWEZ6Xx6oiIfKYR080iu613x5IyBSj4elHIXAAwXMk+SlrKC47VEhNEbkw+9IgIVbm3riYihFQ9czKSB5MksvellYQrP9TsK9DZe0kzxo7BadkDSgji3HroYcDIpIgIVbm3riYihFQ9czKSB5MksvellYQrP9TsK9DZe0kweWD4xiVwmeM+D6BA48nIJs03xD6aQVig5o+ofqoJhmsmFivjwArYRHEB3QVpuM3vggwh9g3AzYmpTLr8uwZ8J5PlGPSQROb/2EmerPhnM852KpwArzUqjj8qnIU+IJnx2m2cIuScrTf9bn5kIkoppBjIEkvaPTvlHoUdWt09SE1p2DDO5jolvRyiBIh4pWq5RxuKcrXgaifl/dAFyf8kqjRMba424ZtSf4yCZErnExDfpAzeIcxbWXo3rQslV1dmV6M6PvgFBHuQVIXV7qWw9JLd1ohke2Wgml78++2zI0g==" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=-6", "result": "x=-2", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{-6}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=\\frac{-6}{3}", "result": "x=-2", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$\\frac{-6}{3}:{\\quad}-2$$", "input": "\\frac{-6}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{6}{3}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{3}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TmmWD/bB/hQAMyRg96RmxS061ljBSPJeENOw2efoSWvmnWoUV5J5qJ5JncpNUfjFo3oe/oyhMy2+1TQhDBd2f2zM6E3fuZxF1XkKAYaRXCD+ASszo4lUparNfNtwUvTCJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "result": "x=-2" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=-2" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=-2" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeIERT4eg56iZ4ajgsLpAp9ko3oe/oyhMy2+1TQhDBd2f2eADB+dAKOx4BAaVcUG+vkD1WOHaLAbMOr6c56Q4EkU9Xn5ws7BVZBR3o6DxbKrOSAArYEP7zrTGWxw0mDr9UE=" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{x^{2}-9}{3x+6}:{\\quad}$$None", "input": "\\frac{x^{2}-9}{3x+6}", "steps": [ { "type": "definition", "title": "Horizontal asymptotes of rational functions", "text": "If denominator's degree > numerator's degree, the x-axis is the horizontal asymptote.<br/>If the degrees are equal, there is an horizontal asymptote: $$y=\\frac{\\mathrm{numerator's\\:leading\\:coefficient}}{\\mathrm{denominator's\\:leading\\:coefficient}}$$<br/>Otherwise, there is no horizontal asymptote." }, { "type": "step", "primary": "The degree of the numerator$$=2.\\:$$The degree of the denominator$$=1$$", "secondary": [ "Numerator's degree > denominator's degree" ] }, { "type": "step", "primary": "Therefore there is no horizontal asymptote" }, { "type": "step", "result": "\\mathrm{No\\:horizontal\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MvDELXwgbpPn4TspOaFeLpHWFYzcA4oaZR79v7JdxE/FjEnStUIOCBezKqcGh3V47Bl54axjF89+pySUHsh45mRtVD7NYSvGjzBssT7OrEbgxO61Or0aV4+NJpNdj06MuA=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{x^{2}-9}{3x+6}:{\\quad}y=\\frac{1}{3}x-\\frac{2}{3}$$", "input": "\\frac{x^{2}-9}{3x+6}", "steps": [ { "type": "definition", "title": "Slant asymptotes of rational functions", "text": "If numerator's degree = 1 + denominator's degree, there is a slant asymptote of the form: y=mx+b.<br/>Otherwise there is no slant asymptote" }, { "type": "step", "primary": "The degree of the numerator$$=2.\\:$$The degree of the denominator$$=1$$", "secondary": [ "Numerator's degree = 1 + denominator's degree, the asymptote is a slant asymptote of the form: $$y=mx+b$$" ] }, { "type": "step", "primary": "For a rational function the slant asymptote is the quotient of the polynomial division" }, { "type": "interim", "title": "Long division $$\\frac{x^{2}-9}{3x+6}:{\\quad}$$Quotient$$=\\frac{x}{3}-\\frac{2}{3},\\:$$Remainder$$=-5$$", "input": "\\frac{x^{2}-9}{3x+6}", "steps": [ { "type": "interim", "title": "Divide $$\\frac{x^{2}-9}{3x+6}:{\\quad}\\frac{x^{2}-9}{3x+6}=\\frac{x}{3}+\\frac{-2x-9}{3x+6}$$", "result": "=\\frac{x}{3}+\\frac{-2x-9}{3x+6}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$x^{2}-9$$<br/>and the divisor $$3x+6\\::\\:\\frac{x^{2}}{3x}=\\frac{x}{3}$$", "result": "\\mathrm{Quotient}=\\frac{x}{3}" }, { "type": "step", "primary": "Multiply $$3x+6$$ by $$\\frac{x}{3}:\\:x^{2}+2x$$", "secondary": [ "Subtract $$x^{2}+2x$$ from $$x^{2}-9$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-2x-9" }, { "type": "step", "primary": "Therefore", "result": "\\frac{x^{2}-9}{3x+6}=\\frac{x}{3}+\\frac{-2x-9}{3x+6}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-2x-9}{3x+6}:{\\quad}\\frac{-2x-9}{3x+6}=-\\frac{2}{3}+\\frac{-5}{3x+6}$$", "result": "=\\frac{x}{3}-\\frac{2}{3}+\\frac{-5}{3x+6}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-2x-9$$<br/>and the divisor $$3x+6\\::\\:\\frac{-2x}{3x}=-\\frac{2}{3}$$", "result": "\\mathrm{Quotient}=-\\frac{2}{3}" }, { "type": "step", "primary": "Multiply $$3x+6$$ by $$-\\frac{2}{3}:\\:-2x-4$$", "secondary": [ "Subtract $$-2x-4$$ from $$-2x-9$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-5" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-2x-9}{3x+6}=-\\frac{2}{3}+\\frac{-5}{3x+6}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "step", "primary": "Simplify", "result": "=\\frac{x}{3}-\\frac{2}{3}-\\frac{5}{3x+6}" } ], "meta": { "solvingClass": "Long Division", "interimType": "Algebraic Manipulation Long Division Title 1Eq" } }, { "type": "step", "primary": "Therefore the slant asymptote is:", "result": "y=\\frac{1}{3}x-\\frac{2}{3}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KL5mikwnGs7EvIxpFPRJ09tfnhM/1hLtN+bYURyIwXQVgQUxJPyUNnGfVirkcwpVO9texQWP30zBG2LsG2smZ5LBuGK/MPjtR4jogh8jhqHIijY9bm25ILUOho7bXci9yiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=-2,\\:\\mathrm{Slant}:\\:y=\\frac{1}{3}x-\\frac{2}{3}" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{x^{2}-9}{3x+6}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }