Linear vs. Asymptotic Graphs

[marcosma]

Hello!

I know that this question has been asked in some capacity a few times on this forum, but I have a slight change to it.

The question from BR Section IV (Periodic Motion) is as follows:

Which graph BEST represents the relationship between the frequency and wavelength of waves emitted from a stereo speaker, if the wave speed is fixed?

I've been able to narrow it down to the two inversely proportional graphs, but I'm having some trouble figuring out asymptotic vs. linear. I eventually understand that it is asymptotic because of f = v(λ^-1), but this leaves me wondering what relationship would ever lead to a negatively sloped, linear graph? The intercepts with 0 seem pretty unlikely to accurately describe any real-life relationships, right? Could someone provide me with an example where a graph that is negative, linear, and intercepting the x- and y-axes? Also, what would the accompanying formula look like for this relationship?

Thanks!

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[Lawpy]

There are two types of inverse relationships: negative and asymptotic. This is because in inverse relations, increasing an independent variable results in decreasing the dependent variable.

An asymptotic relationship is of the form: y = (c / x)^n, where c and n are constants. Usually, n = 1 so you'll get y = c / x. You can see here that increasing the value of x results in y decreasing.

A negative linear relationship is of the form: y = b - mx, where b and m are constant. Note this is similar to the y = mx + b form for positive linear relationships, but here the slope m is negative. Increasing x here will make y more negative and thus decreases the value of y.

Note that if you combine both negative linear and asymptotic relationships, you'll have a direct relationship (although it won't be linear). Combining the two formulas above, you'll get something like y = b - m*(c/x)^n. This is a bit difficult to see so we can simplify things by letting b = 0, m = 1, c = 1 and n = 1 to get y = -1/x. When you increase x, you can see y is becoming less negative and actually increases (this is because -1/5 > -1/4 > -1/3 > -1/2 > -1). The same is true in the more general form.

Another way to distinguish between asymptotic and negative relationships is to see how the dependent variable changes when the independent variable becomes very small. If the independent variable approaches 0 and the dependent variable shoots off to very large numbers and reaches infinity, you have an asymptotic relationship. If the dependent variable instead reaches a relatively normal finite value, you have a negative relationship.

To keep things simple, when you want an asymptotic relationship, just think of reciprocals and fractions, where the independent variable is in the denominator and the numerator is a constant. When you want a negative relationship, just add a negative sign in front of the independent variable. It's a useful shortcut to keep in mind especially when you are dealing with multiple independent variables affecting a single dependent variable.

 

[marcosma]

That makes a lot of sense, thank you!