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Let's analyze the three functions based on the data provided and determine whether each is linear, quadratic, or exponential:

### Function 1:
\[
\begin{array}{c|c}
x & y \\
\hline
4 & 4 \\
5 & -5 \\
6 & 4 \\
7 & 31 \\
8 & 76 \\
\end{array}
\]
- **Check if it's linear**: In a linear function, the difference between consecutive \(y\)-values (called the first differences) should be constant.
  - From \(x = 4\) to \(x = 5\), \( y \) changes from 4 to -5 (difference of -9).
  - From \(x = 5\) to \(x = 6\), \( y \) changes from -5 to 4 (difference of +9).
  - From \(x = 6\) to \(x = 7\), \( y \) changes from 4 to 31 (difference of +27).
  - From \(x = 7\) to \(x = 8\), \( y \) changes from 31 to 76 (difference of +45).
  
  Since the first differences are not constant, **it's not linear**.
  
- **Check for quadratic**: A quadratic function has constant second differences.
  - First differences: \(-9, 9, 27, 45\)
  - Second differences: \(9 - (-9) = 18\), \(27 - 9 = 18\), \(45 - 27 = 18\)
  
  Since the second differences are constant (18), **Function 1 is quadratic**.

### Function 2:
\[
\begin{array}{c|c}
x & y \\
\hline
0 & -3 \\
1 & -10 \\
2 & -21 \\
3 & -36 \\
4 & -55 \\
\end{array}
\]
- **Check if it's linear**:
  - First differences: \(-10 - (-3) = -7\), \(-21 - (-10) = -11\), \(-36 - (-21) = -15\), \(-55 - (-36) = -19\)
  
  The first differences are not constant, so **it's not linear**.

- **Check for quadratic**:
  - First differences: \(-7, -11, -15, -19\)
  - Second differences: \(-11 - (-7) = -4\), \(-15 - (-11) = -4\), \(-19 - (-15) = -4\)
  
  The second differences are constant (-4), so **Function 2 is quadratic**.

### Function 3:
\[
\begin{array}{c|c}
x & y \\
\hline
5 & -5 \\
6 & -9 \\
7 & -13 \\
8 & -17 \\
9 & -21 \\
\end{array}
\]
- **Check if it's linear**:
  - First differences: \(-9 - (-5) = -4\), \(-13 - (-9) = -4\), \(-17 - (-13) = -4\), \(-21 - (-17) = -4\)
  
  Since the first differences are constant, **Function 3 is linear**.

### Final Answers:
- **Function 1**: Quadratic
- **Function 2**: Quadratic
- **Function 3**: Linear

Would you like any further explanations or details on these functions? 

Here are five related questions to further explore this topic:
1. How do you derive a quadratic equation from given points?
2. What are the key characteristics of exponential functions compared to linear and quadratic ones?
3. Can a function be neither linear, quadratic, nor exponential?
4. How do the first and second differences help in identifying quadratic functions?
5. What happens to the rate of change in exponential functions?

**Tip**: To identify exponential functions, check if the ratio of successive \(y\)-values is constant. This indicates a multiplicative pattern rather than an additive one.

For each function, state whether it is linear, quadratic, or exponential.

Identify if given functions are linear, quadratic, or exponential by analyzing tables of values. This problem involves examining first and second differences and checking for constant ratios in exponential functions. Suitable for students in Grades 8-10.

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