Answer
Quartic: $5$ points
Quintic: $6$ points
Work Step by Step
A quartic function van be written in the form:
$$f(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0.$$
In order to determine it, we need to find the values of the constants $a_0,a_1, a_2,a_3, a_4$, therefore we need $5$ points $(x_k,y_k)$ so that for each point we have an equation $y_k=f(x_k)$.
A quintic function van be written in the form:
$$f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0.$$
In order to determine it, we need to find the values of the constants $a_0,a_1, a_2,a_3, a_4,a_5$, therefore we need $6$ points $(x_k,y_k)$ so that for each point we have an equation $y_k=f(x_k)$.
Generalisation: For a polynomial of degree $n$ we need $n+1$ points because we have to determine $n+1$ constants.
