Fitting a Polynomial to Data

Suppose that we have

Table 1: Fitting a polynomial to data.

x	f(x)
3.2	22.0
2.7	17.8
1.0	14.2
4.8	38.3
5.6	51.7

• First, we need to select the points that determine our polynomial. (The maximum degree of the polynomial is always one less than the number of points.) Suppose we choose the first four points.
• If the cubic is $ax^3 + bx^2 + cx + d$, we can write four equations involving the unknown coefficients $a, b, c$, and $d$;
  
  $$\begin{array}{r} when ~x = 3.2: a(3.2)^3 + b(3.2)^2 + c(3.2... ... x=4.8: a(4.8)^3 + b(4.8)^2 + c(4.8) + d = 38.3 \\ \end{array}$$
  
  

• Solving these equations gives
  
  $$\begin{array}{lr} a=& -0.5275 \\ b=& 6.4952 \\ c=& -16.1177 \\ d= & 24.3499 \\ \end{array}$$
  
  
  and our polynomial is
  
  $$-0.5275x^3+6.4952x^2-16.1177x+24.3499$$
  
  
  At $x=3.0$, the estimated value is 20.212.
• if we want a new polynomial that is also made to fit at the point $(5.6,51.7)$ ?
• or if we want to see what difference it would make to use a quadratic instead of a cubic?