The graph of a polynomial function of degree n is completely determined by any n + 1 points on the curve

Question

Highschool maths student here,

I'm having trouble understanding this statement:

"The graph of a polynomial function of degree n is completely determined by any n + 1 points on the curve"

Any help is appreciated, thank you!

Answer 1

This may not be the most mathematically sound answer, being a high schooler myself, but from what I have understood of the statement, the basic premise is that for any polynomial of a given degree n, the graph can be shown with a number of points, n+1. An example of this is with the function y=x+1. Given the points (1,2) and (2,3), we can determine the shape of the graph to be a line which intersects these two points. For higher degrees, the same applies. A quadratic function, with a degree of 2, needs 3 points to be defined ; the vertex and two other points on its curve. I hope this helped, and if I am wrong in any place here feel free to correct me.

Answer 2

a polynomial function of degree n : $\sum_{i=0}^n a_ix^i$

the n+1 coefficients $a_i$ can only be determined by n+1 equations in n+1 points