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Exponential Growth

  1. Aug 29, 2011 #1
    Hi,

    I am explaining the function of each parameter in my model:

    a*bcx

    Are these correct?

    I said that as a changes the graph is stretched parallel to the y axis.

    The movement takes place when we change b and c but I said the growth is exponential... Is that the correct term? Because, for example, if double the value of b the output is quadrupled.

    Thanks,
    Peter G.
     
  2. jcsd
  3. Aug 29, 2011 #2

    Mark44

    Staff: Mentor

    Relative to the graph of y = bcx, a*bcx will be expanded away from the x-axis if a > 1, or compressed toward the x-axis, if 0 < a < 1. If a < 0, there is also a reflection across the x-axis.
    More generally, if y = f(x), the graph of y = af(x) is as explained above. The graph of y = f(cx) will be compressed toward the y-axis, if c > 1, and expanded away from the y-axis, if 0 < c < 1. If c < 0, there is a reflection across the y-axis.
     
  4. Aug 29, 2011 #3
    Hi,

    I was using this graphing program that allows me to use a slider to shift the graph. I was trying to increase the value of a and as I increased it, it moved towards the x axis, eventually even crossing it.

    I agree with the transformations for c you mentioned (do they apply for b too?) but I still don't know if saying that it is compressed or expanded exponentially is correct.

    Thanks once again
     
  5. Aug 29, 2011 #4

    Mark44

    Staff: Mentor

    I don't see how this can happen. What function were you graphing? The transformations I was talking about aren't shifts: they are called stretches or compressions. A shift (or translation) is where you move the graph left or right or up or down.

    Unless there's a vertical translation involved, an exponential function cannot cross the horizontal axis.
    b is the base of your exponential function, so what I said doesn't apply. I didn't say compressed/expanded exponentially. You should omit that word in what you're doing.

    What I said before about the graph of y = f(cx) is correct.

    For example, if y = f(x) = [itex]\sqrt{x}[/itex], the graph of f(2x) is a compression toward the y-axis by a factor of 2. The point (1, 1) on the original graph is now at (1/2, 1), and similar for all other points.

    The graph of y = f(x/3) is a stretch away from the y-axis by a factor of 3. The point (4, 2) on the original graph is now at (12, 2).
     
  6. Aug 29, 2011 #5
    Ok, I got it now. But for the value of a, check this out:

    Let's consider the equation: 2^x.

    When x = 2, y = 4.
    Now, if we plot:

    2*2^x

    When x = 2, y = 8

    So, don't you agree that the graph would increase more rapidly, therefore be stretched parallel to the y-axis? Be compressed towards the x axis.

    (P.S: I'm not trying to argue with you, I know you are a far, far better mathematician than I am :redface: but I don't know, this seems to make sense to me!)
     
    Last edited: Aug 29, 2011
  7. Aug 29, 2011 #6

    Mark44

    Staff: Mentor

    No, I don't agree, but I can see why you're thinking as you are. For your example, relative to the graph of y = 2^x, each y value on the graph of y = 2* 2^x is now doubled, hence all of the points are twice as far away from the x-axis. So to get the graph of y = 2*2^x, we are expanding the points on y = 2^x away from the x-axis by a factor of 2.

    An example that is easier to see is the equation y = 3*sin(x). Each point on the graph of the base function, y = sin(x) is now 3 times as far from the x-axis. The graph of y = 3*sin(x) has been stretched away from (expanded away from) the x-axis by a factor of 3.

    Nit: 2^x is not an equation - it's a function. The equation would be y = 2^x.


     
  8. Aug 30, 2011 #7

    NascentOxygen

    User Avatar

    Staff: Mentor

    You might find it informative to plot log(a*bcx) versus x and examine that as you change the parameters. The graph will be a straight line (not usually horizontal). If will have a defined slope, and a vertical offset, etc., all directly related to the parameters you are discussing. It is an easy way to fit a curve to your raw data.
     
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