Question

Determine the rate of change of the linear function given the table below. Show your work.
x: -3, -2, -1, 0, 1. Y -10, -7, -4, -1, 2.

Answer 1

To determine the rate of change of the linear function represented by the given data in the table, we can use the formula for the rate of change (or slope) between two points. The slope \( m \) is calculated as:

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

We can take any two points from the table to calculate the rate of change. Let's use the first and the last points for clarity:

- Point 1: \( (x_1, y_1) = (-3, -10) \)
- Point 2: \( (x_2, y_2) = (1, 2) \)

Now, substituting these values into the slope formula:

\[
m = \frac{2 - (-10)}{1 - (-3)}
\]

Calculating the numerator:

\[
2 - (-10) = 2 + 10 = 12
\]

Calculating the denominator:

\[
1 - (-3) = 1 + 3 = 4
\]

Now we can find the slope:

\[
m = \frac{12}{4} = 3
\]

Thus, the rate of change of the linear function, or the slope, is \( 3 \).

To ensure that this rate of change is consistent, we can check other pairs of points:

1. Between points \( (-2, -7) \) and \( (-1, -4) \):

\[
m = \frac{-4 - (-7)}{-1 - (-2)} = \frac{-4 + 7}{-1 + 2} = \frac{3}{1} = 3
\]

2. Between points \( (-1, -4) \) and \( (0, -1) \):

\[
m = \frac{-1 - (-4)}{0 - (-1)} = \frac{-1 + 4}{0 + 1} = \frac{3}{1} = 3
\]

3. Between points \( (0, -1) \) and \( (1, 2) \):

\[
m = \frac{2 - (-1)}{1 - 0} = \frac{2 + 1}{1} = \frac{3}{1} = 3
\]

In each case, we consistently find that the rate of change (slope) is \( 3 \).

Thus, the final answer is that the rate of change of the linear function is \( \boxed{3} \).