This is a simple activity to illustrate the concept of polynomial interpolation. The goal is to find a polynomial equation of fourth degree that passes through five points. The coefficients of the interpolating polynomial are controlled by sliders, but the input boxes can also be used for more precise control. The points will change color when the curve passes close to them. You can easily new examples by moving the points straight up or down.
Discussion Questions:
1. Every polynomial of degree 4 or less be expressed as f(x) = a + bx + cx(x-1) + dx(x-1)(x-2) + ex(x-1)(x-2)(x-3). How do we know this?
2. You should discover an easy way to get the curve to pass through all of the points. Can you explain why the method works?
3. Can you think of a way to solve for a, b, c, d, e using algebra instead of trial and error?
4. Why might it be a bad idea to use a fourth-degree polynomial to extrapolate beyond the given data?
5. Why do small changes in e produce such large changes in the graph?
6. Can you generalize to polynomials of higher degree?
Exercise: Let p(x) be a polynomial function of degree 3 such that p(0) = 1, p(1) = -1, p(2) = -5, p(3) = 1, and p(4) = 29. Find a formula for p(x).
