asymptotes of f(x)=(2x^2-9x-5)/(x-5)

{ "query": { "display": "asymptotes $$f\\left(x\\right)=\\frac{2x^{2}-9x-5}{x-5}$$", "symbolab_question": "FUNCTION#asymptotes f(x)=\\frac{2x^{2}-9x-5}{x-5}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "asymptotes", "default": "\\mathrm{Slant}: y=2x+1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Asymptotes of $$\\frac{2x^{2}-9x-5}{x-5}:\\quad\\:$$Slant$$:\\:y=2x+1$$", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$\\frac{2x^{2}-9x-5}{x-5}:{\\quad}$$None", "input": "\\frac{2x^{2}-9x-5}{x-5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x^{2}-9x-5}{x-5}:{\\quad}2x+1$$", "input": "\\frac{2x^{2}-9x-5}{x-5}", "steps": [ { "type": "interim", "title": "Factor $$2x^{2}-9x-5:{\\quad}\\left(2x+1\\right)\\left(x-5\\right)$$", "input": "2x^{2}-9x-5", "result": "=\\frac{\\left(2x+1\\right)\\left(x-5\\right)}{x-5}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2x^{2}-9x-5", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=2,\\:b=-9,\\:c=-5$$", "$$u*v=-10,\\:u+v=-9$$" ] }, { "type": "interim", "title": "Factors of $$10:{\\quad}1,\\:2,\\:5,\\:10$$", "input": "10", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$10:{\\quad}2,\\:5$$", "input": "10", "steps": [ { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRphdXsXWRa5VuJJCXjkRONmBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKkavJ+4fkE2eZqED6K/FpGXNiunDvEaxn8n7nBifSeyk" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$10\\:$$ itself", "result": "1,\\:10" }, { "type": "step", "primary": "The factors of $$10$$", "result": "1,\\:2,\\:5,\\:10" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$10:{\\quad}-1,\\:-2,\\:-5,\\:-10$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-5,\\:-10" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-10,\\:$$check if $$u+v=-9$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-10:\\quad\\:u*v=-10,\\:u+v=-9\\quad\\Rightarrow\\quad\\:$$True", "secondary": [ "Check $$u=2,\\:v=-5:\\quad\\:u*v=-10,\\:u+v=-3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=5,\\:v=-2:\\quad\\:u*v=-10,\\:u+v=3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=10,\\:v=-1:\\quad\\:u*v=-10,\\:u+v=9\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=-10" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2x^{2}+x\\right)+\\left(-10x-5\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwnai4W/KJqpQhPZyuG0nOngrnvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI8Xg/TvH1FSC+/reoRNC3c8f6p27ICA3UoX9R39sy2C3BE8uwPJldwY8ynhYzKxL5odxiol7oWpl+ok4aaIaYadn1fBl+YMO9MWjD6xr3iW9bCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\left(2x^{2}+x\\right)+\\left(-10x-5\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$2x^{2}+x:\\quad\\:x\\left(2x+1\\right)$$", "input": "2x^{2}+x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=2xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(2x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-5\\:$$from $$-10x-5:\\quad\\:-5\\left(2x+1\\right)$$", "input": "-10x-5", "steps": [ { "type": "step", "primary": "Rewrite $$10$$ as $$5\\cdot\\:2$$", "result": "=-5\\cdot\\:2x-5" }, { "type": "step", "primary": "Factor out common term $$-5$$", "result": "=-5\\left(2x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=x\\left(2x+1\\right)-5\\left(2x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$2x+1$$", "result": "=\\left(2x+1\\right)\\left(x-5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x-5$$", "result": "=2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Vertical asymptotes of polynomial functions", "text": "Polynomial functions of degree 2 or higher do not have an asymptote" }, { "type": "step", "result": "\\mathrm{No\\:vertical\\:asymptotes}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dVnUL0bpHaJzHH2MFgu3oeKVttSPlSdrm2mLaU8cWbx6a+NGNBOegbL4QZOpJqJ+t+dQ4rSYgXG4Mug1NPamfwL1VVhs24tHxtUwheCwNQfaYBE59dFawm2eDPOqSND7YsEqQhozMCF6tjrLtoWdzVI=" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$\\frac{2x^{2}-9x-5}{x-5}:{\\quad}$$None", "input": "\\frac{2x^{2}-9x-5}{x-5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x^{2}-9x-5}{x-5}:{\\quad}2x+1$$", "input": "\\frac{2x^{2}-9x-5}{x-5}", "steps": [ { "type": "interim", "title": "Factor $$2x^{2}-9x-5:{\\quad}\\left(2x+1\\right)\\left(x-5\\right)$$", "input": "2x^{2}-9x-5", "result": "=\\frac{\\left(2x+1\\right)\\left(x-5\\right)}{x-5}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2x^{2}-9x-5", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=2,\\:b=-9,\\:c=-5$$", "$$u*v=-10,\\:u+v=-9$$" ] }, { "type": "interim", "title": "Factors of $$10:{\\quad}1,\\:2,\\:5,\\:10$$", "input": "10", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$10:{\\quad}2,\\:5$$", "input": "10", "steps": [ { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRphdXsXWRa5VuJJCXjkRONmBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKkavJ+4fkE2eZqED6K/FpGXNiunDvEaxn8n7nBifSeyk" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$10\\:$$ itself", "result": "1,\\:10" }, { "type": "step", "primary": "The factors of $$10$$", "result": "1,\\:2,\\:5,\\:10" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$10:{\\quad}-1,\\:-2,\\:-5,\\:-10$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-5,\\:-10" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-10,\\:$$check if $$u+v=-9$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-10:\\quad\\:u*v=-10,\\:u+v=-9\\quad\\Rightarrow\\quad\\:$$True", "secondary": [ "Check $$u=2,\\:v=-5:\\quad\\:u*v=-10,\\:u+v=-3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=5,\\:v=-2:\\quad\\:u*v=-10,\\:u+v=3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=10,\\:v=-1:\\quad\\:u*v=-10,\\:u+v=9\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=-10" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2x^{2}+x\\right)+\\left(-10x-5\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwnai4W/KJqpQhPZyuG0nOngrnvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI8Xg/TvH1FSC+/reoRNC3c8f6p27ICA3UoX9R39sy2C3BE8uwPJldwY8ynhYzKxL5odxiol7oWpl+ok4aaIaYadn1fBl+YMO9MWjD6xr3iW9bCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\left(2x^{2}+x\\right)+\\left(-10x-5\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$2x^{2}+x:\\quad\\:x\\left(2x+1\\right)$$", "input": "2x^{2}+x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=2xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(2x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-5\\:$$from $$-10x-5:\\quad\\:-5\\left(2x+1\\right)$$", "input": "-10x-5", "steps": [ { "type": "step", "primary": "Rewrite $$10$$ as $$5\\cdot\\:2$$", "result": "=-5\\cdot\\:2x-5" }, { "type": "step", "primary": "Factor out common term $$-5$$", "result": "=-5\\left(2x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=x\\left(2x+1\\right)-5\\left(2x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$2x+1$$", "result": "=\\left(2x+1\\right)\\left(x-5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x-5$$", "result": "=2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "definition", "title": "Horizontal asymptotes of polynomial functions", "text": "Polynomial functions of degree 1 or higher do not have an horizontal asymptote" }, { "type": "step", "result": "\\mathrm{No\\:horizontal\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0PihooH8V6rf2M5+iqpeDLe57MvsV05FFEFVnvgWNs8SHLz2pBhlzGJ8my2DrfdDmFmXBcNhAKtELLAr4wtrtdEHRDzueUnWP1DanG5XhRZ9uA1nWbwIin+1njlyebafYODrtPBVBqcsD2jO2ci7aPSfX1gkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Slant Asymptotes of $$\\frac{2x^{2}-9x-5}{x-5}:{\\quad}y=2x+1$$", "input": "\\frac{2x^{2}-9x-5}{x-5}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x^{2}-9x-5}{x-5}:{\\quad}2x+1$$", "input": "\\frac{2x^{2}-9x-5}{x-5}", "steps": [ { "type": "interim", "title": "Factor $$2x^{2}-9x-5:{\\quad}\\left(2x+1\\right)\\left(x-5\\right)$$", "input": "2x^{2}-9x-5", "result": "=\\frac{\\left(2x+1\\right)\\left(x-5\\right)}{x-5}", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "2x^{2}-9x-5", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=2,\\:b=-9,\\:c=-5$$", "$$u*v=-10,\\:u+v=-9$$" ] }, { "type": "interim", "title": "Factors of $$10:{\\quad}1,\\:2,\\:5,\\:10$$", "input": "10", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$10:{\\quad}2,\\:5$$", "input": "10", "steps": [ { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:5" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRphdXsXWRa5VuJJCXjkRONmBBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKkavJ+4fkE2eZqED6K/FpGXNiunDvEaxn8n7nBifSeyk" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "2,\\:5" }, { "type": "step", "primary": "Add 1 and the number $$10\\:$$ itself", "result": "1,\\:10" }, { "type": "step", "primary": "The factors of $$10$$", "result": "1,\\:2,\\:5,\\:10" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$10:{\\quad}-1,\\:-2,\\:-5,\\:-10$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-2,\\:-5,\\:-10" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-10,\\:$$check if $$u+v=-9$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-10:\\quad\\:u*v=-10,\\:u+v=-9\\quad\\Rightarrow\\quad\\:$$True", "secondary": [ "Check $$u=2,\\:v=-5:\\quad\\:u*v=-10,\\:u+v=-3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=5,\\:v=-2:\\quad\\:u*v=-10,\\:u+v=3\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=10,\\:v=-1:\\quad\\:u*v=-10,\\:u+v=9\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=1,\\:v=-10" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(2x^{2}+x\\right)+\\left(-10x-5\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwnai4W/KJqpQhPZyuG0nOngrnvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI8Xg/TvH1FSC+/reoRNC3c8f6p27ICA3UoX9R39sy2C3BE8uwPJldwY8ynhYzKxL5odxiol7oWpl+ok4aaIaYadn1fBl+YMO9MWjD6xr3iW9bCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\left(2x^{2}+x\\right)+\\left(-10x-5\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$2x^{2}+x:\\quad\\:x\\left(2x+1\\right)$$", "input": "2x^{2}+x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=2xx+x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(2x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "interim", "title": "Factor out $$-5\\:$$from $$-10x-5:\\quad\\:-5\\left(2x+1\\right)$$", "input": "-10x-5", "steps": [ { "type": "step", "primary": "Rewrite $$10$$ as $$5\\cdot\\:2$$", "result": "=-5\\cdot\\:2x-5" }, { "type": "step", "primary": "Factor out common term $$-5$$", "result": "=-5\\left(2x+1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=x\\left(2x+1\\right)-5\\left(2x+1\\right)" }, { "type": "step", "primary": "Factor out common term $$2x+1$$", "result": "=\\left(2x+1\\right)\\left(x-5\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$x-5$$", "result": "=2x+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "primary": "Slant asymptotes of a linear function is the function itself." }, { "type": "step", "primary": "Therefore the slant asymptote is:", "result": "y=2x+1" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KGMi33OHHY/rSfQuuka/CjFYOWj63HQgxbtNyN59VxfYo3oe/oyhMy2+1TQhDBd2f98zweQxBHifRh6V36jCT3mcEsbsJq/PjahQlnCA3bf07FdORRRBVZ74FjbPEhy89hzC2fiv6R+mr2d6PHCgvDQ=" } }, { "type": "step", "result": "\\mathrm{Slant}:\\:y=2x+1" } ], "meta": { "solvingClass": "Function Asymptotes" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\frac{2x^{2}-9x-5}{x-5}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }