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The table provides values of \(x\) and their corresponding values of \(f(x)\). We need to derive the equation for \(f(x)\) based on the pattern observed in the table.

The table is as follows:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 5 \\
1 & 6 \\
2 & 9 \\
3 & 14 \\
4 & 21 \\
\hline
\end{array}
\]

To find the equation of the function, we will look for a linear or quadratic pattern. Let's start by calculating the differences between consecutive values of \(f(x)\):

\[
\begin{aligned}
f(1) - f(0) &= 6 - 5 = 1, \\
f(2) - f(1) &= 9 - 6 = 3, \\
f(3) - f(2) &= 14 - 9 = 5, \\
f(4) - f(3) &= 21 - 14 = 7.
\end{aligned}
\]

The first differences are not constant, but the second differences (i.e., the differences of the differences) are constant:

\[
\begin{aligned}
3 - 1 &= 2, \\
5 - 3 &= 2, \\
7 - 5 &= 2.
\end{aligned}
\]

This suggests that the function \(f(x)\) is quadratic, and can be written in the form:

\[
f(x) = ax^2 + bx + c
\]

We will now use three points from the table to set up a system of equations and solve for \(a\), \(b\), and \(c\). Using the points \((0, 5)\), \((1, 6)\), and \((2, 9)\), we get the following system:

1. \(f(0) = 5 \Rightarrow a(0)^2 + b(0) + c = 5 \Rightarrow c = 5\)
2. \(f(1) = 6 \Rightarrow a(1)^2 + b(1) + 5 = 6 \Rightarrow a + b + 5 = 6 \Rightarrow a + b = 1\)
3. \(f(2) = 9 \Rightarrow a(2)^2 + b(2) + 5 = 9 \Rightarrow 4a + 2b + 5 = 9 \Rightarrow 4a + 2b = 4 \Rightarrow 2a + b = 2\)

Now, solve this system:

From equation \(a + b = 1\), we get \(b = 1 - a\). Substituting this into \(2a + b = 2\):

\[
2a + (1 - a) = 2 \Rightarrow 2a + 1 - a = 2 \Rightarrow a = 1.
\]

Substituting \(a = 1\) into \(a + b = 1\):

\[
1 + b = 1 \Rightarrow b = 0.
\]

Thus, the equation of the function is:

\[
f(x) = x^2 + 5.
\]

### Final Answer:
\[
f(x) = x^2 + 5.
\]

Would you like more details on any step? 

Here are 5 related questions to extend your understanding:
1. What happens if the second differences are not constant? How do we approach the function then?
2. Can you verify the function by substituting other values from the table?
3. How would you find the equation if the data suggested a linear relationship?
4. Can this method be applied to functions of higher degrees? If yes, how?
5. How can you visually represent this quadratic function on a graph?

**Tip:** For any quadratic function, the second differences between function values will always be constant.

Write the equation for f(x) given the table.

Learn how to find the quadratic function f(x) from a given table of values. This step-by-step guide demonstrates how to identify patterns, use the difference of sequences, and solve for constants in the quadratic formula. Suitable for students in Grades 8-10.

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