1. The problem involves analyzing a curve that passes through points and describes a specific behavior between those points. 2. Key points given are approximately at $x=-1$, $y\lesssim0$, at $(0, -4)$, and at $(2, -7)$. 3. The curve is descending, meaning the function values decrease as $x$ increases from $-1$ to $2$. 4. This suggests the function $f(x)$ is decreasing, so $f'(x)<0$ on this interval. 5. To understand the function better, if we try a quadratic form $f(x)=ax^2+bx+c$, we can use points to find $a, b, c$. 6. Using points: $f(0)=-4 \Rightarrow c=-4$. 7. Using $f(2)=-7$: $4a + 2b -4 = -7$ yields $4a + 2b = -3$. 8. Using $f(-1) \approx 0$: $a - b - 4 \approx 0$ gives $a - b = 4$. 9. Solving the system: $$ a - b = 4 \\ 4a + 2b = -3 $$ Multiply first by 2: $2a - 2b=8$, add to second: $4a+2b=-3$ gives $6a=5$, so $a=\frac{5}{6}$. 10. Substitute back: $\frac{5}{6} - b=4 \Rightarrow b= \frac{5}{6} - 4 = -\frac{19}{6}$. 11. Function: $f(x) = \frac{5}{6}x^2 - \frac{19}{6}x -4$. 12. The graph shape fits the description: downward curve passing given points. 13. The slope $f'(x) = \frac{5}{3}x - \frac{19}{6}$ is negative between $x=-1$ and $2$, consistent with the problem. Final answer: $$f(x) = \frac{5}{6}x^2 - \frac{19}{6}x - 4$$