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Make a conjecture about y = ax+b

  1. Aug 24, 2013 #1
    1. The problem statement, all variables and given/known data
    I have to make a conjecture about y = ax+b in terms of the ratio of the x coordinate in regards to the y-coordinate of the function.


    2. Relevant equations

    y = ax + b

    3. The attempt at a solution

    So I need to investigate the function like this:

    y = ax + b

    f(x)=x2+y2

    f(x)=x2+(ax+b)(ax+b)

    f(x)=x2+a2 x2+2axb+b2

    But I'm fairly sure need to get a result that the ratio x/y is equal to a/1
    But I usually end up at x/y = 1/a+b
    when using the method I've written above.
     
  2. jcsd
  3. Aug 25, 2013 #2

    tiny-tim

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    Hi 5ymmetrica1! :smile:
    How do you get that? :confused:

    (and what does r2, = x2 + y2, have to do with it anyway??)
     
  4. Aug 25, 2013 #3

    mfb

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    Try to look at (two or maybe more) y-values for different x-values, and see how their differences are.
     
  5. Aug 25, 2013 #4
    because this relates to an earlier problem I posted and is investigated in the same way which was answered with this
    so I have to use this
    f(x) = x2 + y2 where y = ax+b

    so by expanding and differentiating I have tried several ways and seem to always get to 1/ a+b

    but I've noticed that the ratio of the x coordinate/ y coordinate is usually x:1
    and this value of x is in most cases the same as the value of 'a'

    For example in the function y = -2x+5
    the value of x = 2 and y =1
    the ratio is 2:1

    and I checked with my teacher who said that by doing the same with y= ax+b
    I should get to something similar such as a/1

    I hope someone can help me here as I'm completely lost and have been stuck on this problem for nearly a week.
     
  6. Aug 26, 2013 #5

    tiny-tim

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    But that problem asked you to find the position that is closest to the origin (ie to minimise x2 + y2).

    This question doesn't, does it? :confused:

    What exactly is the question?​
     
  7. Aug 28, 2013 #6
    I have to observe what I noticed when investigating the original function and make a conjecture about it.

    Then I'm required to prove this conjecture with algebra.

    The only thing I noticed was that in the ratio of x:y, the number of 'x' was usually equal to 'a' in the equation ax+b

    So for the function -2x+5 has a ratio of 2:1 since the 'a' value is 2
    and for the function -4x+7 has a ratio of 4:1 since the 'a' value is 4
    ect.

    So by proving this with algebra I need to get a result of a:1
     
  8. Aug 28, 2013 #7

    haruspex

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    You mean that it was approximately equal, I hope.
    Under what circumstances (x values) does the accuracy improve? Does this suggest anything about a trend?
     
  9. Aug 29, 2013 #8

    tiny-tim

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    ohh, i think i see what the question is now …

    are you asked to find the slope of the shortest line from the origin to y = ax + b ?
     
  10. Aug 30, 2013 #9
    Yes exactly, but I also need to relate this back to the original function, this question is basically asking.

    What did you notice, make a conjecture about it.
    Then prove this conjecture with algebra.

    To do this I need the slope of the shortest line of y = ax+b
     
  11. Aug 30, 2013 #10

    tiny-tim

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    i've done it (now i understand the question), and i don't get that

    start by differentiating the distance-squared, and finding the value of x for which that is the minimum,

    then put that value of x into ax + b to find y

    then find y/x :wink:

    show us what you get :smile:
     
  12. Aug 31, 2013 #11
    Do you mean like this?


    f(x)=x2+a2 x2+2axb+b2

    f '(x) = 2x + 2a(2x) + 2

    2x + 2a(2x) + 2 = 0

    2x + 2a(2x) = -2

    2x = -2 - 2a(2x)

    x = -1 - ax

    y = a(-1 -ax) + b
     
  13. Aug 31, 2013 #12

    tiny-tim

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    Hi 5ymmetrica1! :smile:

    Yes, exactly like that!

    Except, your algebra is completely up the spout :redface:
    no, that should be 2x + a2(2x) + 2ab :wink:
    nooo !

    try again :smile:
     
  14. Aug 31, 2013 #13
    Thanks Tim, I've been away from math for a while so my algebra is pretty rusty

    so I'm now getting

    2x + a2(2x) +2ab = 0

    2x + a2(2x) = -2ab

    2x = -2ab - a2(2x)

    x = -ab - a2x

    is this correct?
     
  15. Sep 1, 2013 #14

    tiny-tim

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    (just got up :zzz:)

    better! :smile:

    but don't you need x on the left and a constant on the right? :wink:
     
  16. Sep 1, 2013 #15
    So you mean I need all the x's on the left and the constant a^2 on the right?
     
  17. Sep 2, 2013 #16

    tiny-tim

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    a2 ?

    no :confused:
     
  18. Sep 2, 2013 #17
    well I know 'b' was a constant

    or do you mean I need to add a constant 'c' like in anti-differentiation?

    I'm confused aswell :P
     
  19. Sep 3, 2013 #18
    so I now know I need to get a final result of -a, which proves my original hypothesis that the value of the ratio is equal to the value of a in the equation y = ax+b

    and it involves using the equation √x2 + a2x2+2axb+b2

    but i'm stuck from here. Can anyone offer me any help here. This assignment is due tomorrow and I'm still stuck on this problem!
     
  20. Sep 3, 2013 #19

    CompuChip

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    If the letters confuse you, maybe it's useful setting the constants to a numeric value (e.g. a = 2, b = 3) and first working out the answer with those example values. I.e. maybe try solving
    $$x = -2 \cdot 3 - 2^2 x$$
    first.
     
  21. Sep 3, 2013 #20

    vela

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    You were close here. As Tim said, you need to get the x's all on one side and the constants on the other. Only the red term doesn't depend on x. It's the constant, not a2, you want on the righthand side.

    Go back to your second equation (the one in blue) because it has all the terms with x on the lefthand side and the constant on the righthand side. Divide out the common factor of 2, and then try solve for x again.
     
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