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write an equation for the functions on this table

To find an equation for the function represented by the table, let's first observe the pattern in the values of \( f(x) \) as \( x \) increases.

Given the table:

\[
\begin{array}{c|c}
x & f(x) \\
\hline
2 & -1 \\
3 & 0 \\
4 & 3 \\
5 & 8 \\
6 & 15 \\
\end{array}
\]

### Step 1: Observe the pattern

The differences between consecutive values of \( f(x) \) are:

\[
\begin{array}{c|c}
x & f(x) & \text{First Differences} \\
\hline
2 & -1 & \\
3 & 0  & 0 - (-1) = 1 \\
4 & 3  & 3 - 0 = 3 \\
5 & 8  & 8 - 3 = 5 \\
6 & 15 & 15 - 8 = 7 \\
\end{array}
\]

The first differences themselves are increasing. This suggests the function might be quadratic since the first differences of a quadratic function increase linearly.

### Step 2: Check for a quadratic function

A general quadratic function has the form:
\[
f(x) = ax^2 + bx + c
\]

We need to find the coefficients \( a \), \( b \), and \( c \). Let's use the three points \( (2, -1) \), \( (3, 0) \), and \( (4, 3) \) to set up a system of equations.

1. For \( x = 2 \):
   \[
   4a + 2b + c = -1 \quad \text{(Equation 1)}
   \]
2. For \( x = 3 \):
   \[
   9a + 3b + c = 0 \quad \text{(Equation 2)}
   \]
3. For \( x = 4 \):
   \[
   16a + 4b + c = 3 \quad \text{(Equation 3)}
   \]

### Step 3: Solve the system of equations

Subtract Equation 1 from Equation 2:
\[
(9a + 3b + c) - (4a + 2b + c) = 0 - (-1)
\]
\[
5a + b = 1 \quad \text{(Equation 4)}
\]

Subtract Equation 2 from Equation 3:
\[
(16a + 4b + c) - (9a + 3b + c) = 3 - 0
\]
\[
7a + b = 3 \quad \text{(Equation 5)}
\]

Subtract Equation 4 from Equation 5:
\[
(7a + b) - (5a + b) = 3 - 1
\]
\[
2a = 2 \quad \Rightarrow a = 1
\]

Substitute \( a = 1 \) into Equation 4:
\[
5(1) + b = 1 \quad \Rightarrow b = -4
\]

Substitute \( a = 1 \) and \( b = -4 \) into Equation 1:
\[
4(1) + 2(-4) + c = -1
\]
\[
4 - 8 + c = -1 \quad \Rightarrow c = 3
\]

### Final Equation:
So the function is:
\[
f(x) = x^2 - 4x + 3
\]

### Verification:
Let's check this function with the values in the table:

- For \( x = 2 \): \( f(2) = 2^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \) ✔
- For \( x = 3 \): \( f(3) = 3^2 - 4(3) + 3 = 9 - 12 + 3 = 0 \) ✔
- For \( x = 4 \): \( f(4) = 4^2 - 4(4) + 3 = 16 - 16 + 3 = 3 \) ✔
- For \( x = 5 \): \( f(5) = 5^2 - 4(5) + 3 = 25 - 20 + 3 = 8 \) ✔
- For \( x = 6 \): \( f(6) = 6^2 - 4(6) + 3 = 36 - 24 + 3 = 15 \) ✔

The function \( f(x) = x^2 - 4x + 3 \) is indeed the correct equation for the given table.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How can we determine if a function is quadratic by looking at its table of values?
2. What is the general process for finding a quadratic function given three points?
3. How do the first and second differences relate to the degree of a polynomial?
4. Can this method be used for higher-degree polynomials, and if so, how?
5. What are some real-world scenarios where quadratic functions are used?

**Tip:** Always verify your derived function by plugging back in the values from the original table to ensure accuracy.

Learn how to write an equation for the functions represented by a given table using a step-by-step solution. This problem involves identifying patterns in the values, setting up a system of equations, and solving for the coefficients of a quadratic function. Suitable for high school students.

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