The problem asks us to determine which equation defines the quadratic function $f(x)$, given the table of $x$ values and corresponding $f(x)$ values:

The options for the quadratic function are:

• A) $f(x) = 3x^2 + 3x + 14$
• B) $f(x) = 5x^2 + x + 14$
• C) $f(x) = 9x^2 - x + 14$
• D) $f(x) = x^2 + 5x + 14$

Step 1: Substitute $x = -1$, $x = 0$, and $x = 1$ into each equation to see if the function matches the corresponding $f(x)$ values.

Option A: $f(x) = 3x^2 + 3x + 14$

• For $x = -1$: $f(-1) = 3(-1)^2 + 3(-1) + 14 = 3(1) - 3 + 14 = 14$ But $f(-1)$ should be 10. This does not match.

Option B: $f(x) = 5x^2 + x + 14$

• For $x = -1$: $f(-1) = 5(-1)^2 + (-1) + 14 = 5(1) - 1 + 14 = 18$ This is not 10, so Option B is incorrect.

Option C: $f(x) = 9x^2 - x + 14$

• For $x = -1$: $f(-1) = 9(-1)^2 - (-1) + 14 = 9(1) + 1 + 14 = 24$ This also doesn't match $f(-1) = 10$.

Option D: $f(x) = x^2 + 5x + 14$

• For $x = -1$: $f(-1) = (-1)^2 + 5(-1) + 14 = 1 - 5 + 14 = 10$ This matches $f(-1) = 10$.

• For $x = 0$: $f(0) = (0)^2 + 5(0) + 14 = 0 + 0 + 14 = 14$ This matches $f(0) = 14$.

• For $x = 1$: $f(1) = (1)^2 + 5(1) + 14 = 1 + 5 + 14 = 20$ This matches $f(1) = 20$.

Since Option D satisfies all the values in the table, the correct equation is: $\boxed{f(x) = x^2 + 5x + 14}$

Would you like further details on solving quadratic functions?

Related Questions:

1. How do you derive the equation of a quadratic function given data points?
2. What is the standard form of a quadratic equation?
3. How do transformations affect the graph of a quadratic function?
4. Can quadratic functions model real-world scenarios like projectile motion?
5. How can you find the vertex and axis of symmetry of a quadratic function?

Tip: Always check multiple points when matching a function to a set of data values!