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Homework Help: How do I sketch the graph y= (2x+5)/(x-1)

  1. Feb 10, 2010 #1
    How do I sketch the graph y= (2x+5)/(x-1)
     
  2. jcsd
  3. Feb 10, 2010 #2

    Mentallic

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    Homework Helper

    Re: Hyperbola

    Say you want to simplify
    [tex]2+\frac{3}{2x+1}[/tex]
    then you would multiply the first constant by the highest common denominator so you can put it all in one fraction as so:
    [tex]\frac{2(2x+1)+3}{2x+1}[/tex]
    and then simplify and you get
    [tex]\frac{4x+5}{2x+1}[/tex].
    But what if you started with this expression and wanted to go back to the start now?

    You need to cancel out the x in the numerator so you need to convert it into the form
    [tex]\frac{a(2x+1)+b}{2x+1}[/tex]
    Notice that the numerator has 4x in it so a=2 so that we can have this 4x.
    [tex]\frac{2(2x+1)+b}{2x+1}[/tex]
    Now in order to keep the numerator the same we need a number b so that it makes [itex]2(2x+1)+b=4x+5[/itex] and once you solve this you get b=3.
    Then we get
    [tex]\frac{2(2x+1)+3}{2x+1}=\frac{2(2x+1)}{2x+1}+\frac{3}{2x+1}=2+\frac{3}{2x+1}[/tex]

    From this expression we can easily sketch the graph
    [tex]y=2+\frac{3}{2x+1}[/tex]
    by noting that there is a vertical asymptote at 2x+1=0, x=-1/2 and there is a horizontal asymptote at y=2 (since the fraction never equals 0).

    Now see if you can apply the same idea to your problem.
     
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