asymptotes of f(x)= 4/((x-2)^2)

{ "query": { "display": "asymptotes $$f\\left(x\\right)=\\frac{4}{\\left(x-2\\right)^{2}}$$", "symbolab_question": "FUNCTION#asymptotes f(x)=\\frac{4}{(x-2)^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Vertical}: x=2,\\mathrm{Horizontal}: y=0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{4}{\\left(x-2\\right)^{2}}:\\quad\\:$$Vertical$$:\\:x=2,\\:$$Horizontal$$:\\:y=0$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{4}{\\left(x-2\\right)^{2}}:{\\quad}x=2$$", "input": "\\frac{4}{\\left(x-2\\right)^{2}}", "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 $$\\frac{4}{\\left(x-2\\right)^{2}}$$ and compare to zero" }, { "type": "interim", "title": "Solve $$\\left(x-2\\right)^{2}=0:{\\quad}x=2$$", "input": "\\left(x-2\\right)^{2}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x-2=0:{\\quad}x=2$$", "input": "x-2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "x-2=0", "result": "x=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "x-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "x=2" } ], "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 solution to the quadratic equation is:", "result": "x=2" } ], "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/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKgPjQHEHt5n6ch9SLpoHkaRSpN33oxZMojoqvYhvSJAMBxyAX3aEQ7z5W7QbMWGScOIBjVD0w41lRvWu8QFmOAP5QAXWBZpSggry8iSlkmxBZFQ8OchBdgxMOJrEMyk3s=" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{4}{\\left(x-2\\right)^{2}}:{\\quad}y=0$$", "input": "\\frac{4}{\\left(x-2\\right)^{2}}", "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$$=0.\\:$$The degree of the denominator$$=2$$", "secondary": [ "Denominator's degree > numerator's degree. Therefore, the horizontal asymptote is the x-axis" ] }, { "type": "step", "primary": "The horizontal asymptote is:", "result": "y=0" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MvOdg7ltJNuuiR1KMjwHNbNZEt3ZXAiqUE0HIXrrrezJO7tpg6+wppGIR16yRwSoZwhY1KWFY4dVpjknbwHn5uV1KblWOvna2Z9Rkh1r3jbdi0pB2BMd8KF7v7124UkMFc=" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{4}{\\left(x-2\\right)^{2}}:{\\quad}$$None", "input": "\\frac{4}{\\left(x-2\\right)^{2}}", "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$$=0.\\:$$The degree of the denominator$$=2$$", "secondary": [ "Numerator's degree $$\\neq\\:$$ 1 + denominator's degree" ] }, { "type": "step", "primary": "Therefore there is no slant asymptote" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KFYYtUbK2TosA7yAb9UyxIoZaVNqMPeX7r6qpcoccVGW8LfSxJ+0AgVLpCSnLX0iSgI5i9maKPYxQTldBPZ3WKEDxf53IU3WN8riM9ddB/uwBaTln1cW+tqpqZFH0GoZCw==" } }, { "type": "step", "result": "\\mathrm{Vertical}:\\:x=2,\\:\\mathrm{Horizontal}:\\:y=0" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{4}{(x-2)^{2}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }