Science:Math Exam Resources/Courses/MATH307/December 2010/Question 03 (d)/Statement

Suppose we are given 4 points (x₁, y₁), (x₂, y₂), (x₃, y₃) and (x₄, y₄) in the plane and we want to find a function ƒ(x), defined for $x_1\le x\le x_4$, whose graph interpolates these points. Assume that

$f(x)=\{\begin{array}{ll}{p}_1(x)& \text{for }{x}_1\le x\le x_2\\ p_2(x)& \text{for }{x}_2\le x\le x_3\\ p_3(x)& \text{for }{x}_3\le x\le x_4\end{array}$

where each p_i(x) is a polynomial.

(d) When each p_i(x) is a cubic polynomial of the form

$a_i(x-x_1{)}^3+b_i(x-x_i{)}^2+c_i(x-x_i)+d_i$

the equations written in parts (a), (b) and (c) above are equivalent to a system of linear equations in the unknowns a_i, b_i, c_i and d_i, i = 1, 2, 3. How many more equations are needed if there are to be the same number of equations as unknowns? What equations are usually added and why?