An Interpolation problem.

Definition: A function f is said to interpolate the point [x1,y1] if f(x1) = y1. This is a fancy way of saying the graph of f passes through the point [x1,y1].

Problem. Find the line which interpolates the points [0,4] and [1,6].

Solution. Here f(x) = a*x + b. The equations are f(0) = b = 4 and f(1) = a + b = 6. So a = 6 - b = 2. The line is f(x) = 2*x + 4.

Problem. Find the quadratic polynomial which interpolates [2,0], [6,1], and [8,0]

Solution start. Let . The the equations are {f(2)=0, f(6)=1, f(8)=0}. This is 3 linear equations in the 3 unknowns a,b and c.

Solve this by hand and/or with maple.

Problem. (This is the one I gave in class last Thursday) Find the polynomial of which interpolates (ie, passes through) the points [2,3],[0,2],[5,7],[-12,14],[7,13],[a,10]

Solution. Set up the polynomial function. The coefficients c1 through c6 are the unknowns.

> f := x->c1+c2*x + c3*x^2 + c4*x^3 +c5*x^4 + c6*x^5 ;

Set up the equations that the polynomial must satisfy. There are 4 values given, so there are 4 equations.

> data:=[ [2,3],[0,2],[5,7],[-12,14],[7,13],[a,10]];

> eqns := [seq(f(data[i][1])=data[i][2],i=1..nops(data))];

Convert to a matrix equation. (This is not really necessary, but is one way to do it. See the alternative below.)

alternative