asymptotes of (2x^2)/(x+3)

{ "query": { "display": "asymptotes $$\\frac{2x^{2}}{x+3}$$", "symbolab_question": "FUNCTION#asymptotes \\frac{2x^{2}}{x+3}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=-3,\\mathrm{Slant}: y=2x-6", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{2x^{2}}{x+3}:\\quad\\:$$Vertical$$:\\:x=-3,\\:$$Slant$$:\\:y=2x-6$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{2x^{2}}{x+3}:{\\quad}x=-3$$", "input": "\\frac{2x^{2}}{x+3}", "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:$${\\quad}x=-3$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$\\frac{2x^{2}}{x+3}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$x+3=0:{\\quad}x=-3$$", "input": "x+3=0", "steps": [ { "type": "interim", "title": "Move $$3\\:$$to the right side", "input": "x+3=0", "result": "x=-3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "x+3-3=0-3" }, { "type": "step", "primary": "Simplify", "result": "x=-3" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The following points are undefined", "result": "x=-3" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "The vertical asymptotes are:", "result": "x=-3" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKSvJfPyQRlLyxIvNJXs5dG72wZm7kDUxdE6YSmfEbr2sklkbd4Y9e4UaOc4Z+Fp+6sEuKHoiRQqchnbg8AZByYO+Iqy4t/faUR9WuCBCer7rCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{2x^{2}}{x+3}:{\\quad}$$None", "input": "\\frac{2x^{2}}{x+3}", "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+iqpeDLe57MsZs2IoCSYfurSSJo1bQcV8o3oe/oyhMy2+1TQhDBd2fzDpSTjniHqaFLbyPyboRgNODRvwVMhMzO4XxkPsuS18Vuu4jWPr/0i9JbffB93ZA/o2Xaqjwnh4zwzPT+vbWpo=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{2x^{2}}{x+3}:{\\quad}y=2x-6$$", "input": "\\frac{2x^{2}}{x+3}", "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 $$\\frac{2x^{2}}{x+3}:{\\quad}$$Quotient$$=2x-6,\\:$$Remainder$$=18$$", "input": "\\frac{2x^{2}}{x+3}", "steps": [ { "type": "interim", "title": "Divide $$\\frac{2x^{2}}{x+3}:{\\quad}\\frac{2x^{2}}{x+3}=2x+\\frac{-6x}{x+3}$$", "result": "=2x+\\frac{-6x}{x+3}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$2x^{2}$$<br/>and the divisor $$x+3\\::\\:\\frac{2x^{2}}{x}=2x$$", "result": "\\mathrm{Quotient}=2x" }, { "type": "step", "primary": "Multiply $$x+3$$ by $$2x:\\:2x^{2}+6x$$", "secondary": [ "Subtract $$2x^{2}+6x$$ from $$2x^{2}$$ to get new remainder" ], "result": "\\mathrm{Remainder}=-6x" }, { "type": "step", "primary": "Therefore", "result": "\\frac{2x^{2}}{x+3}=2x+\\frac{-6x}{x+3}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } }, { "type": "interim", "title": "Divide $$\\frac{-6x}{x+3}:{\\quad}\\frac{-6x}{x+3}=-6+\\frac{18}{x+3}$$", "result": "=2x-6+\\frac{18}{x+3}", "steps": [ { "type": "step", "primary": "Divide the leading coefficients of the numerator $$-6x$$<br/>and the divisor $$x+3\\::\\:\\frac{-6x}{x}=-6$$", "result": "\\mathrm{Quotient}=-6" }, { "type": "step", "primary": "Multiply $$x+3$$ by $$-6:\\:-6x-18$$", "secondary": [ "Subtract $$-6x-18$$ from $$-6x$$ to get new remainder" ], "result": "\\mathrm{Remainder}=18" }, { "type": "step", "primary": "Therefore", "result": "\\frac{-6x}{x+3}=-6+\\frac{18}{x+3}" } ], "meta": { "interimType": "PolyDiv Subtract Divide 1Eq" } } ], "meta": { "solvingClass": "Long Division", "interimType": "Algebraic Manipulation Long Division Title 1Eq" } }, { "type": "step", "primary": "Therefore the slant asymptote is:", "result": "y=2x-6" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KGMi33OHHY/rSfQuuka/CjFohrPRnzu8TplCSFMJzsVc1sD7NfhsPe7eDHrmjY0mE0aSv0vc8ePHy0ar7mWvNf6Qc04bfKnfo9Nm04R2a152EgxUX6HyvcEVWJuW9+07pg==" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=-3,\\:\\mathrm{Slant}:\\:y=2x-6" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{2x^{2}}{x+3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }