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In the image provided, the task is to determine whether the given tables of values represent a quadratic function. There are four tables labeled as (4), (3), (2), and (1), with `x` and `y` values listed.

### To check if the tables represent a quadratic function:
A quadratic function will have a constant second difference between the `y` values when the `x` values are equally spaced. Here's the process:

1. **Find the first differences** between consecutive `y` values.
2. **Find the second differences** (the difference between consecutive first differences).

If the second differences are constant, the function is quadratic. Let's evaluate each table:

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### Table (4):
- **x values**: -1, 0, 1, 2
- **y values**: 5, 10, 15, 20

#### First differences:
- \( 10 - 5 = 5 \)
- \( 15 - 10 = 5 \)
- \( 20 - 15 = 5 \)

#### Second differences:
- \( 5 - 5 = 0 \)
- \( 5 - 5 = 0 \)

Since the second differences are zero, **this is not a quadratic function**.

---

### Table (3):
- **x values**: -7, -6, -5, -4, -3
- **y values**: 43, 31, 21, 13, 7

#### First differences:
- \( 31 - 43 = -12 \)
- \( 21 - 31 = -10 \)
- \( 13 - 21 = -8 \)
- \( 7 - 13 = -6 \)

#### Second differences:
- \( -10 - (-12) = 2 \)
- \( -8 - (-10) = 2 \)
- \( -6 - (-8) = 2 \)

Since the second differences are constant (2), **this is a quadratic function**.

---

### Table (2):
- **x values**: 0, 1, 2, 3
- **y values**: 30, 124, 342, 728

#### First differences:
- \( 124 - 30 = 94 \)
- \( 342 - 124 = 218 \)
- \( 728 - 342 = 386 \)

#### Second differences:
- \( 218 - 94 = 124 \)
- \( 386 - 218 = 168 \)

Since the second differences are not constant, **this is not a quadratic function**.

---

### Table (1):
- **x values**: -3, -2, -1, 0, 1
- **y values**: 12, 6, 2, 0, 0

#### First differences:
- \( 6 - 12 = -6 \)
- \( 2 - 6 = -4 \)
- \( 0 - 2 = -2 \)
- \( 0 - 0 = 0 \)

#### Second differences:
- \( -4 - (-6) = 2 \)
- \( -2 - (-4) = 2 \)
- \( 0 - (-2) = 2 \)

Since the second differences are constant (2), **this is a quadratic function**.

---

### Summary:
- **Table (4)**: Not quadratic.
- **Table (3)**: Quadratic.
- **Table (2)**: Not quadratic.
- **Table (1)**: Quadratic.

Would you like more details or further explanation on any step?

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Here are some follow-up questions:
1. How can you confirm whether a function is quadratic using its equation?
2. What is the significance of second differences being constant?
3. Can you provide another example of a quadratic table and solve it?
4. How does a cubic function differ from a quadratic function in terms of differences?
5. What are the graphical characteristics of quadratic functions?

**Tip:** Always double-check second differences when determining if a function is quadratic—it’s an essential indicator!

Determine whether the given tables of values represent a quadratic function, and state the reason if not.

Learn how to determine whether a set of values represents a quadratic function by calculating first and second differences. Includes detailed solutions for multiple examples, suitable for students in Grades 9-10.

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