Quadratic Function Analysis

Very Hard Algebra Functions

Let the function $$f(x)$$ be defined by the equation:

$$f(x) = ax^2 + bx + c$$

for some constants $$a$$, $$b$$, and $$c$$ where $$a eq 0$$. If it is known that the vertex of the parabola represented by this function lies at the point $$(2, -1)$$, and the function passes through the point $$(4, 3)$$, which of the following is true regarding the values of $$a$$, $$b$$, and $$c$$?

It is true that $$f(2) = -1$$ and $$f(4) = 3$$.

It is true that the values of $$b$$ and $$c$$ can be expressed in terms of $$a$$.

The values of $$a$$, $$b$$, and $$c$$ are independent of each other.

At least one of the constants $$a$$, $$b$$, or $$c$$ must be positive.

Hint

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