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B Help with understanding Nature of Roots for Quadratic and Cu

  1. Oct 23, 2016 #1
    Hi

    I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that - the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to determine the values of the shift to obtain the required roots. I attach 3 example questions...
    They are 8.1.4, 7.4 and 4.5

    Please help me find Youtube Videos/Websites or any resource that might help me understand how to approach these questions.

    Thanks smile.png

    4.5.png 8.1.4.png 7.4.png
     
  2. jcsd
  3. Oct 24, 2016 #2

    Simon Bridge

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    OK the questions do not ask for exact values for roots so there is not much calculation needed: it's a way of testing your understanding.
    Going: ##g(x)\to g(x)+k## ... and asking about the roots, is the same as asking for the intercept between curve ##y=g(x)## and the line ##y=-k## ... for example, asking what k must be to give the g(x)+k two positive real roots, you are just looking for a horizontal line that cuts g(x) twice... or the set of such lines, and so on depending on the wording.

    ie. ##g(x)=x^2## has 1 real root at x=0, you see why that is right?
    A horizontal line drawn through y>0 will cut g(x) twice - do you see why?
    Therefore, g(x)+k will have two real roots for k<0.

    ie. ##g(x)=(x-1)x^2## has 2 roots, x=0 and x=1 (where are the turning points?)
    A horizontal line drawn for any y<0 will cut g(x) once only, do you see why?
    So k>0 makes g(x)+k have one root, and the root is positive.
    It has three roots for a small range of values - can you see which?
     
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