https://file.aiquickdraw.com/mathbot/file/1728252226398580ap808.jpg

Let's analyze the table and check which equation matches the relationship.

The table provides values for \( x \) and \( y \):

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 3 \\
0 & 2 \\
4 & 0 \\
6 & -1 \\
\hline
\end{array}
\]

We are given four equations to choose from:
- \( A: y = -x + 1 \)
- \( B: y = \frac{x}{2} - 2 \)
- \( C: y = -x + 2 \)
- \( D: y = -\frac{x}{2} + 2 \)

### Step 1: Check Equation A (\( y = -x + 1 \))

- For \( x = -2 \), \( y = -(-2) + 1 = 2 + 1 = 3 \) (Correct)
- For \( x = 0 \), \( y = -(0) + 1 = 1 \) (Incorrect)
  
This is not the correct equation.

### Step 2: Check Equation B (\( y = \frac{x}{2} - 2 \))

- For \( x = -2 \), \( y = \frac{-2}{2} - 2 = -1 - 2 = -3 \) (Incorrect)
  
This is not the correct equation.

### Step 3: Check Equation C (\( y = -x + 2 \))

- For \( x = -2 \), \( y = -(-2) + 2 = 2 + 2 = 4 \) (Incorrect)
  
This is not the correct equation.

### Step 4: Check Equation D (\( y = -\frac{x}{2} + 2 \))

- For \( x = -2 \), \( y = -\frac{-2}{2} + 2 = 1 + 2 = 3 \) (Correct)
- For \( x = 0 \), \( y = -\frac{0}{2} + 2 = 0 + 2 = 2 \) (Correct)
- For \( x = 4 \), \( y = -\frac{4}{2} + 2 = -2 + 2 = 0 \) (Correct)
- For \( x = 6 \), \( y = -\frac{6}{2} + 2 = -3 + 2 = -1 \) (Correct)

### Conclusion:
The correct equation is **D: \( y = -\frac{x}{2} + 2 \)**.

Would you like further details or have any questions?

### Related Questions:
1. How can we derive the slope of a line from a table of values?
2. What is the difference between a positive and negative slope?
3. How can we check the y-intercept of a line using a table?
4. How do you interpret fractional slopes like \( -\frac{1}{2} \)?
5. What are the steps to convert a table of values into a linear equation?

**Tip**: Always test each given equation with all the points in the table to verify which one fits best.

Which equation describes the relation indicated by the table? The table lists four values of x and their corresponding y values: (-2, 3), (0, 2), (4, 0), and (6, -1), and four possible equations: A: y = -x + 1, B: y = x/2 - 2, C: y = -x + 2, D: y = -x/2 + 2.

Learn how to determine the equation that best describes the relationship between x and y based on a given table of values. This problem involves algebraic concepts like linear equations and the slope-intercept form. Suitable for Grades 8-10.

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