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To determine the rate of change of a linear function represented by a table of values, we need to calculate the change in \( y \) for each unit change in \( x \).

Let's look at consecutive points in the table:

1. From \( x = -4 \) to \( x = -2 \):
   - \( y \) changes from \( -1 \) to \( 0 \).
   - Change in \( y \) ( \( \Delta y \) ) = \( 0 - (-1) = 1 \).
   - Change in \( x \) ( \( \Delta x \) ) = \( -2 - (-4) = 2 \).
   - Rate of change = \( \frac{\Delta y}{\Delta x} = \frac{1}{2} \).

2. From \( x = -2 \) to \( x = 0 \):
   - \( y \) changes from \( 0 \) to \( 1 \).
   - Change in \( y \) = \( 1 - 0 = 1 \).
   - Change in \( x \) = \( 0 - (-2) = 2 \).
   - Rate of change = \( \frac{1}{2} \).

3. From \( x = 0 \) to \( x = 2 \):
   - \( y \) changes from \( 1 \) to \( 2 \).
   - Change in \( y \) = \( 2 - 1 = 1 \).
   - Change in \( x \) = \( 2 - 0 = 2 \).
   - Rate of change = \( \frac{1}{2} \).

Since the rate of change is consistent at \( \frac{1}{2} \) between each consecutive pair of points, we can conclude that the rate of change for this linear function is \( \frac{1}{2} \).

Would you like more details, or do you have any questions?

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Here are 5 related questions that expand on this topic:

1. How would the rate of change differ if the function were non-linear?
2. Can you write the equation of the line for this function?
3. What is the significance of the rate of change in a real-world context?
4. How can we identify the slope of a function from its graph?
5. What does a negative rate of change indicate?

**Tip:** The rate of change, or slope, is a key characteristic in identifying linearity in a function. For linear functions, the rate of change is constant across the domain.

The table below represents a linear function. Identify the rate of change of the function.

Learn how to identify the rate of change of a linear function using a table of values. This example includes steps to calculate the slope using changes in x and y values, reinforcing key concepts in algebra for Grades 7-9.

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