For distinct \(x_1,\ldots,x_n\) and arbitrary \(y_1,\ldots,y_n\) there exists a unique interpolating polynomial \(p_{n-1}(x)\) passing through the points \((x_1,y_1),\ldots,(x_n,y_n)\), shown as the grey curve below:
Drag the red points with your finger/mouse!
Making small changes to a point causes large oscillations far away from the point.
Typically polynomial interpolation gives a curve with large oscillations near the endpoints, and NOT a "nice smooth curve we would expect" (cubic spline, shown as blue curve below).