numpy.polyfit example

 #%% load package

import numpy as np

import matplotlib.pyplot as plt

#%% example 1: simple second order fitting

# assume y=2x**2

xx = np.arange(-3,3,step=0.2)

yy = 2*xx**2 

yy +=  (np.random.random(len(xx))-0.5)  # add noise

plt.figure(1, dpi=150)

plt.plot(xx, yy, 'ro')

# curve fitting

lin_fit = np.polyfit(xx, yy, 2)

print(lin_fit)

x2 = np.arange(-3,3,step=.01)

y2 = np.polyval(lin_fit, x2)

plt.figure(1)

plt.plot(x2,y2, 'k')

plt.legend(['raw data','fitting curve'])

plt.legend(['raw data','fitting curve'])

plt.plot(x2[np.argmin(y2)], np.min(y2), 'b*', markersize=10)

plt.text(x2[np.argmin(y2)], np.min(y2)-1, 

         'min: (%.2f, %.2f)'%(x2[np.argmin(y2)], np.min(y2)))

plt.axis([-3,3,-2,20])

plt.show()

#=================================================

#%% example 2

# assume y-y0 = a*(x-x0)**2

# --> y = ax**2 - 2a*x0*x + a*x0**2 + y0

# let a=-1, x0=1.2, y0=0.7, i.e. the maximum value is located at (1.2, 0.7)

# --> y = -1x**2 + 2.4x - 0.74

xx = np.arange(-3,3,step=0.2)

yy = -1*xx**2 + 2.4*xx - 0.74 

yy +=  (np.random.random(len(xx))-0.5)  # add noise

plt.figure(2, dpi=150)

plt.plot(xx, yy, 'ro')

# curve fitting

lin_fit = np.polyfit(xx, yy, 2)

print(lin_fit)

x2 = np.arange(-3,3,step=.01)

y2 = np.polyval(lin_fit, x2)

plt.figure(2)

plt.plot(x2,y2, 'k')

plt.legend(['raw data','fitting curve'])

plt.plot(x2[np.argmax(y2)], np.max(y2), 'b*', markersize=10)

plt.text(x2[np.argmax(y2)], np.max(y2)+1, 

         'max: (%.2f, %.2f)'%(x2[np.argmax(y2)], np.max(y2)))

plt.axis([-3,3,-20,3])

plt.show()

==================================

It's noted that the 

x0 = lin_fit[1] / (-2*lin_fit[0])

y0 = y0 = lin_fit[2] - lin_fit[0] * x0**2

https://github.com/Renfong/Python-toolbox/blob/main/np_polyfit_example.py