1. The problem involves understanding the function values given in the table for $x$ from $-3$ to $5$ and the corresponding $y$ values. 2. The table shows $x$ values: $-3, -2, -1, 0, 1, 2, 3, 4, 5$. 3. The $y$ values are: $7, 0, -5, -8, -9, -8, -5, 0, 7$. 4. We observe the pattern of $y$ values is symmetric around $x=1$ and $x=4$ with minimum at $x=1$ where $y=-9$. 5. The function appears to be quadratic or related to quadratic expressions, as indicated by the presence of $x^2$, $-2x^2$, and $-8$ in the table. 6. To find the function rule, consider the quadratic form $y = ax^2 + bx + c$. 7. Using points $(0,-8)$, $(1,-9)$, and $(2,-8)$ to find $a$, $b$, and $c$: - At $x=0$: $y = c = -8$ - At $x=1$: $a(1)^2 + b(1) + c = -9 \Rightarrow a + b - 8 = -9 \Rightarrow a + b = -1$ - At $x=2$: $4a + 2b - 8 = -8 \Rightarrow 4a + 2b = 0$ 8. Solve the system: From $a + b = -1$, multiply by 2: $2a + 2b = -2$ Subtract from $4a + 2b = 0$: $(4a + 2b) - (2a + 2b) = 0 - (-2) \Rightarrow 2a = 2 \Rightarrow a = 1$ Then $b = -1 - a = -1 - 1 = -2$ 9. The function is $y = x^2 - 2x - 8$. 10. Verify with $x=3$: $y = 9 - 6 - 8 = -5$, matches the table. 11. Final answer: $$y = x^2 - 2x - 8$$