Question

Find the parabola of the form y=ax² b which best fits the points (1,0),(4,4),(5,8) by minimizing the sum, s, of squares of the vertical distances from the above points to the parabola given by s=(a b)² (16a b−4)² (25a b−8)² .y=​

Answer 1

The parabola that best fits the given points (1,0), (4,4), and (5,8) by minimizing the sum of squares of the vertical distances is y = 11/2 * x² - 25/2 * x + 12.

Step-by-step explanation:

To find the parabola that minimizes the sum of squares of the vertical distances, we start with the given sum s = (ab)²(16ab - 4)²(25ab - 8)². Our goal is to minimize this expression by determining the values of a and b. We can use the given points (1,0), (4,4), and (5,8) to form three equations.

Let's denote the points as (x₁, y₁), (x₂, y₂), and (x₃, y₃) respectively. The equations become:

y₁ = ax₁² + bx₁

y₂ = ax₂² + bx₂

y₃ = ax₃² + bx₃

Substituting the given points, we get a system of three equations. Solving this system, we find the values of a and b. The resulting parabola is y = 11/2 * x² - 25/2 * x + 12, which provides the best fit for the given points.

This parabola minimizes the sum of squares of the vertical distances from the points to the curve. The process involves careful substitution and solving, ensuring that the resulting parabola is a precise fit for the specified data points.