Make a parabola steeper at both sides while keeping both ends

May 25, 2022

I’m having a parabola with both axes being from 0 to 1 as follows:

The parabola is created and normalized with the following code:

import matplotlib.pyplot as plt
import numpy as np

# normalize array
def min_max_scale_array(arr):
    arr = np.array(arr)
    return (arr - arr.min())/(arr.max()-arr.min())

x = np.linspace(-50,50,100)
y = x**2

x = min_max_scale_array(x)
y = min_max_scale_array(y)

fig, ax = plt.subplots()
ax.plot(x, y)

I want to create another one with both ends being the same but both sides become steeper like this:

I thought of joining an exponential curve and its reflection but that would make the resulting parabola looks pointy at the bottom.

Can you show me how to achieve this? Thank you!

>Solution :

If you want to modify any arbitrary curve, you can change the x values, for example taking a power of it:

# x and y are defined

for factor in [1.1, 1.5, 2, 3, 4]:
    x2 = 2*x-1
    x3 = (abs(x2)**(1/factor))*np.sign(x2)/2+0.5
    ax.plot(x3, y, label=f'{factor=}')

output: