To create a quadratic function that best fits the curve passing through the given points, we can use the method of least squares or directly set up the quadratic function as:

$f(x) = ax^2 + bx + c$

We have the following points:

$(0.2, 0.063), \, (0.4, 0.167), \, (0.6, 0.313), \, (0.8, 0.52), \, (1, 1)$

These points will give us a system of equations when plugged into the quadratic form $f(x) = ax^2 + bx + c$. Let's substitute each point into the general form and solve the resulting system:

Step 1: Form equations using the points

For $x = 0.2, y = 0.063$: $a(0.2)^2 + b(0.2) + c = 0.063$ $0.04a + 0.2b + c = 0.063 \quad \text{(Equation 1)}$

For $x = 0.4, y = 0.167$: $a(0.4)^2 + b(0.4) + c = 0.167$ $0.16a + 0.4b + c = 0.167 \quad \text{(Equation 2)}$

For $x = 0.6, y = 0.313$: $a(0.6)^2 + b(0.6) + c = 0.313$ $0.36a + 0.6b + c = 0.313 \quad \text{(Equation 3)}$

For $x = 0.8, y = 0.52$: $a(0.8)^2 + b(0.8) + c = 0.52$ $0.64a + 0.8b + c = 0.52 \quad \text{(Equation 4)}$

For $x = 1, y = 1$: $a(1)^2 + b(1) + c = 1$ $a + b + c = 1 \quad \text{(Equation 5)}$

Step 2: Solve the system of equations

Now, we need to solve the system of 5 equations with 3 unknowns (a, b, and c). However, since we have more points than necessary for a quadratic (3 points should ideally be enough), we can use methods like least squares to find the best-fitting solution. Let me calculate that for you. numpy as np

Given points

points = np.array([[0.2, 0.063], [0.4, 0.167], [0.6, 0.313], [0.8, 0.52], [1, 1]])

Separate x and y

x = points[:, 0] y = points[:, 1]

Set up the design matrix for quadratic fitting (Ax = b, where A is the design matrix and x = [a, b, c])

A = np.vstack([x**2, x, np.ones_like(x)]).T

Solve for the coefficients using least squares

coeffs = np.linalg.lstsq(A, y, rcond=None)[0]

Coefficients a, b, c

coeffsThe quadratic function that best fits the given points, using the least squares method, is:

$f(x) = 1.4518x^2 - 0.6286x + 0.151$

This function should provide a good estimate of the curve passing through the points $(0.2, 0.063)$, $(0.4, 0.167)$, $(0.6, 0.313)$, $(0.8, 0.52)$, and $(1, 1)$.

Would you like a graph of this quadratic curve along with the points to visually inspect the fit? Or do you have any further questions about quadratic regression?

Additional questions to explore:

1. How do we interpret the coefficients of a quadratic function in real-world applications?
2. What are the common methods for curve fitting besides least squares?
3. How can we find the roots (x-intercepts) of the quadratic equation?
4. How does changing the number of data points affect the fit of the quadratic function?
5. Can a quadratic function always perfectly fit any set of data points?

Tip: When solving a system of equations from multiple points, the least squares method provides an optimal solution that minimizes the total error between the observed values and the predicted ones.