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Graph sketching

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Sketch the graph of the function y(x) = (x-3)/ [(x+1)*(x-2)], indicating the positions of the turning points. Prove that there is a range of values which y can't take if x is real.

    2. Relevant equations

    3. The attempt at a solution

    To draw the graph, I found

    1. the vertical asymptotes which are x = -1 and x = 2.
    2. As x tends to -1 from the left, y tends to -ve infinity.
    As x tends to -1 from the right, y tends to +ve infinity.
    As x tends to 2 from the left, y tends to +ve infinity.
    As x tends to 2 from the right, y tends to -ve infinity.
    3. As x tends to -ve infinity, y tends to 0 from below the x-axis.
    As x tends to +ve infinity, y tends to 0 from above the x-axis.
    4. The turning points are (1,1) and (5,1/9).

    The graph can be drawn using 1-4.

    I think so far I have got everything right. The problem is with proving that there is a range of values which y can't take if x is real.

    I considered the x-axis number line in chunks:

    1. x < -1 : y < 0.
    2. -1 < x < 2 : y > 1.
    3. x > 2 : y < 1/9.

    This shows that 1/9 < y < 1 is not in the range if the domain consists of real x.

    Does this constitute a valid proof?
     
  2. jcsd
  3. Mar 14, 2012 #2
    I agree with all of the work you have done. I would just add a little more substance to your proof, why is y great than or less than those numbers? I would say something about the critical points (or turning points as you call them).
     
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