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Let's work through this step by step.

The table provided gives values for \(x\) and \(y\), which represent points on the line. To find the equation of the linear function, we can calculate the slope \(m\) using the formula for the slope between two points:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

We can take two pairs of points from the table. Let's use (-5, 17) and (-2, 2):

\[
m = \frac{2 - 17}{-2 - (-5)} = \frac{-15}{3} = -5
\]

So, the slope is \(m = -5\).

Next, we can use the slope-intercept form of a line, which is:

\[
y = mx + b
\]

We know \(m = -5\). Now, to find the y-intercept \(b\), we substitute one of the points from the table into the equation. Let's use the point (-2, 2):

\[
2 = -5(-2) + b
\]
\[
2 = 10 + b
\]
\[
b = 2 - 10 = -8
\]

So, the equation of the line is:

\[
y = -5x - 8
\]

The correct answer is **D) \(y = -5x - 8\)**.

Would you like more details or have any questions? 

Here are 5 questions that expand on this:

1. How do you determine the slope if the two points have the same x-coordinate?
2. What happens if the points have the same y-coordinate? How does the slope change?
3. Could we use a different pair of points from the table to verify the slope? 
4. How can you recognize a linear function just by looking at a table of values?
5. What does the slope tell us about the relationship between \(x\) and \(y\) in this function?

**Tip**: When solving linear equations, always check two points to ensure the calculated slope is consistent.

This table of values represents a linear function. Which of the following equations represents the function?

Learn how to determine the equation of a linear function from a table of values. This problem covers concepts like calculating slope and using the slope-intercept form y = mx + b. Suitable for students in Grades 8-10.

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