Answer: D. $f(x) = x^2 + 9x + 12$

Explanation:1. The table shows three values of $x$ and their corresponding values of $f(x)$ for a quadratic function $f$. The points are: - $(-1, 4)$ - $(0, 12)$ - $(1, 22)$

2. We can test each given quadratic equation to see which one is satisfied by all three points.

Option A: $f(x) = 5x^2 + 5x + 12$ - For $x = -1$: $f(-1) = 5(-1)^2 + 5(-1) + 12 = 5(1) - 5 + 12 = 5 - 5 + 12 = 12$. Expected $4$. (Incorrect)

Option B: $f(x) = 9x^2 + x + 12$ - For $x = -1$: $f(-1) = 9(-1)^2 + (-1) + 12 = 9(1) - 1 + 12 = 9 - 1 + 12 = 8 + 12 = 20$. Expected $4$. (Incorrect)

Option C: $f(x) = 9x^2 - x + 12$ - For $x = -1$: $f(-1) = 9(-1)^2 - (-1) + 12 = 9(1) + 1 + 12 = 9 + 1 + 12 = 10 + 12 = 22$. Expected $4$. (Incorrect)

Option D: $f(x) = x^2 + 9x + 12$ - For $x = -1$: $f(-1) = (-1)^2 + 9(-1) + 12 = 1 - 9 + 12 = -8 + 12 = 4$. (Matches) - For $x = 0$: $f(0) = (0)^2 + 9(0) + 12 = 0 + 0 + 12 = 12$. (Matches) - For $x = 1$: $f(1) = (1)^2 + 9(1) + 12 = 1 + 9 + 12 = 10 + 12 = 22$. (Matches) Since Option D satisfies all three points, it is the correct equation.

3. Alternative method: Using the general form $f(x) = ax^2 + bx + c$. - From $(0, 12)$: $f(0) = a(0)^2 + b(0) + c = 12 \implies c = 12$. So, $f(x) = ax^2 + bx + 12$. - From $(-1, 4)$: $f(-1) = a(-1)^2 + b(-1) + 12 = 4 \implies a - b + 12 = 4 \implies a - b = -8$ (Equation 1) - From $(1, 22)$: $f(1) = a(1)^2 + b(1) + 12 = 22 \implies a + b + 12 = 22 \implies a + b = 10$ (Equation 2) - Add Equation 1 and Equation 2: $(a - b) + (a + b) = -8 + 10$ $2a = 2$ $a = 1$ - Substitute $a=1$ into Equation 2: $1 + b = 10 \implies b = 9$. - So, $f(x) = 1x^2 + 9x + 12 = x^2 + 9x + 12$. This matches Option D.

4. Conclusion: The equation that defines $f$ is $f(x) = x^2 + 9x + 12$.