Sometimes we can find a polynomial function that passes through all the data points, but for others we have to limit ourselves to find the polynomial that fits them better. Each adjustment can be done with different methods for different conditions, and each has its own advantages. We will discuss each case one at a time.
Using the (n+1) Point Principle
If the data set consists of (n+1) points, where n is the degree of the polynomial function we want as model, then we can use the (n+1) Point Principle.
Point Principle
For any set of n+1 points in the coordinate plane that pass the vertical line test, there is a unique polynomial of degree at most n that fist the points perfectly.
Therefore, if the given points have different x-coordinates, according to this principle, there will be a polynomial of at most degree n passing through them. We can then try to adjust a polynomial by recalling its standard form.
y=a_nx^n + a_(n-1)x^(n-1)+...+a_1x+a_0
Since we have (n+1) points, we can substitute the x- and y-coordinates of each point in the polynomial function, obtaining (n+1) equations with (n+1) variables that we can solve for the (n+1) parameters a_0, a_1, ..., a_n.
Using a Calculator
This method requires having a calculator at hand, which can be a drawback. However, this method has other benefits.
We can try different polynomial models — usually linear, quadratic, and cubic.
If the data set consists of n+1 points and we adjust for a n degree polynomial, we can obtain a curve fitting them perfectly, just as with the other method, under the same conditions.
For this method we proceed by entering the data into the calculator list L1 and L2. Then, we chose between the options LinReg, QuadReq, or CubicReg. The calculator will perform the regression and provide R^2, which can give us an idea of how good the adjustment is. The closer to 1 the better the fit. An example is shown below.
