How to model two parabolic peaks of different magnitudes and inclines using a single polynomial equation

Question

I recently came upon a graph of two parabolic curves of different magnitudes and intensities and attempted to model it with a single parabolic equation. Rough graph I have the coordinates of the vertices: (3, 667), (8, 1287). I have a few other points as well: (1, 55), (2, 187), (4, 188), (5, 86), (6, 120), (7, 375), (9, 13).

I was wondering if there was a systematic way to go about creating this equation.

My first thought was to transform the graph several units down and find the x-intercepts after doing so, then write the equation in factored form {y = -a(x-x_i)(x-x_ii)...). For instance, I transformed the graph 600 units down, found the x-ints to be 2.86, 3.14, 7.25, 8.75. I used the desmos online calculator and tried a few a-values, settling for a= 15, and by finagling the x-intercept values, could more closely align the graph with the original data points.

The equation I settled on was -15(x-2.56)(x-3.34)(x-6.83)(x-9.27)+600. However, I was wondering if there was a more streamlined way to get to an equation.

Answer 1

My suggestions are as follows:

1. Find more coordinates on the graph
2. Plot the points corresponding to these coordinates on GeoGebra.
3. Use the FitPoly method to find the best polynomial approximation for the given points. This method will computationally minimise the sum of the squares of the residuals to give the best match for the points.

Using the coordinates provided in the question, the equation I got using the above method (for a polynomial of degree $7$) was $f(x) \approx -0.97x^{7}+33.81x^{6}-483.50x^{5}+3643.69x^{4}-15411.56x^{3}+35825.7x^{2}-41137.97x+17585.8$ but there is scope for much better approximations using more points.