1. The problem asks to identify the type of function $f(x)$ based on the given table values: | $x$ | $f(x)$ | |-----|--------| | 1 | 1 | | 2 | 2.5 | | 3 | 3 | | 4 | 2.5 | | 5 | 1 | | 6 | -1.5 | 2. We analyze the pattern of $f(x)$ values to determine if the function is linear or quadratic. 3. A linear function has a constant rate of change (constant slope). Let's check the differences: $$\Delta f = f(x+1) - f(x)$$ Between $x=1$ and $x=2$: $2.5 - 1 = 1.5$ Between $x=2$ and $x=3$: $3 - 2.5 = 0.5$ Between $x=3$ and $x=4$: $2.5 - 3 = -0.5$ Between $x=4$ and $x=5$: $1 - 2.5 = -1.5$ Between $x=5$ and $x=6$: $-1.5 - 1 = -2.5$ 4. The differences are not constant, so $f(x)$ is not linear. 5. Next, check if $f(x)$ is quadratic by examining the second differences: Second differences: Between $1.5$ and $0.5$: $0.5 - 1.5 = -1$ Between $0.5$ and $-0.5$: $-0.5 - 0.5 = -1$ Between $-0.5$ and $-1.5$: $-1.5 - (-0.5) = -1$ Between $-1.5$ and $-2.5$: $-2.5 - (-1.5) = -1$ 6. The second differences are constant and negative ($-1$), indicating a quadratic function with a negative leading coefficient. 7. Therefore, the best description is: **D. Quadratic, with a negative leading coefficient**.