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The table shows the values of \( x \) and their corresponding values of \( y \). It looks like we are trying to identify a relationship between \( x \) and \( y \), possibly through a function or equation. Let's analyze the points:

| \( x \) | \( y \) |
|:--:|:--:|
| -3  |  6  |
| -2  |  2  |
| -1  |  0  |
|  0  |  0  |
|  1  |  2  |
|  2  |  6  |

### Analysis
To find the equation that describes the relationship between \( x \) and \( y \), we can observe that the pattern seems symmetric. This suggests the possibility of a quadratic relationship, so we could try fitting a quadratic equation of the form:
\[
y = ax^2 + bx + c
\]

#### Step 1: Use points to form a system of equations
We can substitute known points into the general quadratic form:

1. For \( x = 0, y = 0 \):
   \[
   0 = a(0)^2 + b(0) + c \implies c = 0
   \]
   So, the equation becomes \( y = ax^2 + bx \).

2. For \( x = 1, y = 2 \):
   \[
   2 = a(1)^2 + b(1) \implies a + b = 2
   \]

3. For \( x = 2, y = 6 \):
   \[
   6 = a(2)^2 + b(2) \implies 4a + 2b = 6
   \]

#### Step 2: Solve the system of equations
Now we have the system:
1. \( a + b = 2 \)
2. \( 4a + 2b = 6 \)

Let's solve this system. From the first equation:
\[
b = 2 - a
\]

Substitute \( b = 2 - a \) into the second equation:
\[
4a + 2(2 - a) = 6
\]
Simplifying:
\[
4a + 4 - 2a = 6 \implies 2a = 2 \implies a = 1
\]

Substitute \( a = 1 \) into \( b = 2 - a \):
\[
b = 2 - 1 = 1
\]

#### Step 3: Write the final equation
The equation that fits the data is:
\[
y = x^2 + x
\]

This quadratic equation seems to model the relationship between \( x \) and \( y \) as observed from the table.

Would you like more details on this process, or do you have any further questions?

### Relative Questions:
1. How can you determine if the relationship between two variables is quadratic?
2. What other methods can be used to find the equation of a function from a table of values?
3. What are the different forms of a quadratic equation, and when are they useful?
4. Can this equation \( y = x^2 + x \) be expressed in factored form, and what would that look like?
5. How does the symmetry of the points suggest that the relationship might be quadratic?

### Tip:
When trying to find a pattern in a table of values, check the differences between consecutive \( y \)-values. If the second differences are constant, the relationship is likely quadratic!

The table shows values of x and y, where y depends on x. What is the relationship between x and y, and what equation can represent this relationship?

Analyze a table of x and y values to determine the quadratic relationship between them. Using algebraic methods, this example demonstrates how to derive the equation y = x^2 + x by solving a system of equations. Suitable for students in Grades 8-10.

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