Values of a, b, and c for a quadratic function, given the x, y intercepts and slope of tangent line?

Question

What are the values of $a$, $b$, and $c$, for the function $f(x)= ax^2 + bx + c$ if $f(x)$ has an $x$-intercept at $x=1$, a $y$-intercept at $y=-2$, and a tangent line $x=0$ with a slope of $m_{\text{tan}} = -1$?

Answer 1

A possible approach:

Note that $y=-2$ at $x=0$ (Implies that $f(0)=-2$). What can you deduce about the value of $c$?

There is a tangent line at $x=0$ with slope $m=-1$ (Implies that $f'(0)=-1$), what does that tell you about the value of $b$?

Note that $f(x)$ has an $x$-intercept at $x=1$ (Implies $f(1)=0$). What does that tell you about the value of $a$?