Final answer:
To find the parabola that best fits the given points, we minimize the sum of squares by finding the values of a and b that make the sum of squares a minimum. This can be done using calculus and setting the derivative of the sum of squares equal to zero. Once we have the values of a and b, we substitute them into the equation y = ax² + b to obtain the parabola of best fit.
Step-by-step explanation:
To find the parabola that best fits the given points, we need to minimize the sum of squares, S = (a+b)² + (9a+b-3)² + (16a+b-6)². This can be done by finding the values of a and b that make S a minimum. We can achieve this by using calculus and setting the derivative of S with respect to a and b equal to zero. Solving these equations will give us the values of a and b, and we can then substitute these values into the equation y = ax² + b to obtain the parabola of best fit.
Let's calculate the derivatives first:
dS/da = 2(a+b) + 2(9a+b-3)(9) + 2(16a+b-6)(16) = 2a+2b+18(9a+b-3) + 32(16a+b-6)
dS/db = 2(a+b) + 2(9a+b-3) + 2(16a+b-6)(6) = 2a+2b+9(9a+b-3) + 6(16a+b-6)
Setting these derivatives equal to zero gives us:
2a+2b+18(9a+b-3) + 32(16a+b-6) = 0
2a+2b+9(9a+b-3) + 6(16a+b-6) = 0
Solving these equations will yield the values of a and b. Once we have these values, we can substitute them into the equation y = ax² + b to obtain the parabola of best fit.