Quadratic Function 96F867

1. **State the problem:** We need to find the quadratic function $f(x)$ that passes through the points $(3, 0)$ and $(-4, 0)$ and has a leading coefficient of $-2$. 2. **Recall the factored form of a quadratic:** If a quadratic has roots $r_1$ and $r_2$, it can be written as $$f(x) = a(x - r_1)(x - r_2)$$ where $a$ is the leading coefficient. 3. **Identify the roots:** The roots given are $3$ and $-4$, so the factors are $(x - 3)$ and $(x + 4)$. 4. **Use the leading coefficient:** The leading coefficient is $-2$, so $$f(x) = -2(x - 3)(x + 4)$$ 5. **Check the other options:** - $-2(x - 4)(x - 3)$ has roots $4$ and $3$, which is incorrect. - $(x - 4)(x + 3)$ has roots $4$ and $-3$, and leading coefficient $1$, incorrect. - $-2(x + 3)(x - 4)$ has roots $-3$ and $4$, incorrect. 6. **Final answer:** $$\boxed{f(x) = -2(x - 3)(x + 4)}$$