Understanding Exponential Growth in a*bcx Model

[Peter G.]

Hi,

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

a*b^cx

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.

 

[Mark44]

Hi,

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

a*b^cx

Are these correct?

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

Relative to the graph of y = b^cx, a*b^cx 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.

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.

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.

 

[Peter G.]

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

 

[Mark44]

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 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.

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.

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) = \sqrt{x}, 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).

 

[Peter G.]

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 but I don't know, this seems to make sense to me!)

 

[Mark44]

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.

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.

(P.S: I'm not trying to argue with you, I know you are a far, far better mathematician than I am but I don't know, this seems to make sense to me!)

 

[NascentOxygen]

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

a*b^cx

You might find it informative to plot log(a*b^cx) 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.