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:

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?

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!