Cotangent Parabolas?!

You may have noticed that the parabolas formed by multiplying the function of cotangent lines results in a downward facing parabola. The slopes being opposites, leading to the coefficient of the quadratic term to be negative.

Is there an upward facing parabola that is tangent to two given cotangent lines? The answer is yes. An example appears below:

Is there a relationship between the functions of these "cotangent parabolas," much like there was between the functions of cotangent lines? To answer this it would be helpful to expand the the function fg:

fg (x) = (2x + 4)(-2x -3)

The second parabola's function is:

From observation, one might hypothesize that the coefficients of the quadratic and linear terms of one function are the opposite of their corresponding coefficients in the second function. Another hypothesis is that, like the y-intercepts of cotangent lines, the y-intercepts these parabolas sum to one.

Testing these hypotheses, a set of cotangent lines are chosen:

f(x) = 3x - 7

g(x) = -3x + 8

The quadratic function produced by their product is:

fg (x) = (3x - 7)(-3x + 8)

Using the hypotheses above, the prediction for the function of the cotantent parabola, h, is:

The graphs of f ,g, fg, and h are below.

The proof of these hypotheses is left to the viewer of this webpage. I would be interested in seeing it. Here is a link to my email.