Suppose $f(x)$ is a polynomial of the form $a x^{2}+b x+c,$ with $a, b, c$ unknown real numbers. Suppose you are additionally told that $f(1)=2$ and $f(-1)=3$. Consider the following four statements.
$\text{(S1)}$ $f(0)$ cannot be determined from the given data.
$\text{(S2)}$ $f(2)$ cannot be determined from the given data.
$\text{(S3)}$ Both $f(0)$ and $f(2)$ can be determined from the given data.
$\text{(S4)}$ $f(0)$ and $f(2)$ satisfy $3 f(0)+f(2)=9$.
Which of the above statements are true?
- Statements $\text{(S1)}$ and $\text{(S2)}$ only
- Statements $\text{(S1), (S2)}$, and $\text{(S4)}$ only
- Statement $\text{(S3)}$ only
- Statements $\text{(S3)}$ and $\text{(S4)}$ only
- All four statements are true
