https://file.aiquickdraw.com/mathbot/file/1727285058823hsu04lgj.jpg

Let's analyze the three functions in the table to determine their types:

### **Function 1**
- \( x = 5 \), \( y = 768 \)
- \( x = 6 \), \( y = 192 \)
- \( x = 7 \), \( y = 48 \)
- \( x = 8 \), \( y = 12 \)
- \( x = 9 \), \( y = 3 \)

Each time \( x \) increases by 1, \( y \) is divided by 4. This shows a **constant multiplicative change**, which is characteristic of an **exponential function**.

### **Function 2**
- \( x = 1 \), \( y = -14 \)
- \( x = 2 \), \( y = -9 \)
- \( x = 3 \), \( y = -6 \)
- \( x = 4 \), \( y = -5 \)
- \( x = 5 \), \( y = -6 \)

Here, the differences between consecutive \( y \)-values are not constant. The first differences are: \( -9 - (-14) = 5 \), \( -6 - (-9) = 3 \), \( -5 - (-6) = 1 \), and so on. Since the second differences are constant, this suggests that the function is **quadratic**.

### **Function 3**
- \( x = 2 \), \( y = -11 \)
- \( x = 3 \), \( y = -13 \)
- \( x = 4 \), \( y = -17 \)
- \( x = 5 \), \( y = -23 \)
- \( x = 6 \), \( y = -31 \)

The differences between consecutive \( y \)-values are: \( -13 - (-11) = -2 \), \( -17 - (-13) = -4 \), \( -23 - (-17) = -6 \), and \( -31 - (-23) = -8 \). Since the differences are constant and linear, this is a **linear function**.

### Conclusion:
1. **Function 1**: Exponential
2. **Function 2**: Quadratic
3. **Function 3**: Linear

Would you like more details or explanations on these function types? Here are five questions to explore:

1. How can you determine a function is quadratic from its table?
2. What is the general form of an exponential function?
3. How do you differentiate between exponential and linear functions using a table of values?
4. What are second differences in relation to quadratic functions?
5. How does knowing the type of function help in graphing it?

**Tip:** Always check for constant differences (for linear), constant second differences (for quadratic), and multiplicative patterns (for exponential) when analyzing tables of values.

Analyze three functions based on their given tables of values and determine whether they are linear, quadratic, exponential, or none of the above.

This problem helps students identify the types of functions (linear, quadratic, or exponential) by analyzing tables of values. The problem includes examples of constant differences, second differences, and multiplicative changes, suitable for students in Grades 9-12.

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