Question

change in y/change in x

x
-4,0,4,8,12

y
6,4,2,0,-2

Answer 1

To find the change in \( y \) over the change in \( x \) (often referred to as the slope), you can use the formula:

\[
\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

We can calculate the slope using different pairs of points from the data you provided. Let’s use the first and last points for a broader perspective:

1. **Using \( (x_1, y_1) = (-4, 6) \) and \( (x_5, y_5) = (12, -2) \)**:

\[
\Delta y = -2 - 6 = -8
\]
\[
\Delta x = 12 - (-4) = 12 + 4 = 16
\]
\[
\text{slope} = \frac{-8}{16} = -\frac{1}{2}
\]

Now, let's check the slope between consecutive points to see if it remains the same:

2. **Between \( (-4, 6) \) and \( (0, 4) \)**:

\[
\Delta y = 4 - 6 = -2
\]
\[
\Delta x = 0 - (-4) = 0 + 4 = 4
\]
\[
\text{slope} = \frac{-2}{4} = -\frac{1}{2}
\]

3. **Between \( (0, 4) \) and \( (4, 2) \)**:

\[
\Delta y = 2 - 4 = -2
\]
\[
\Delta x = 4 - 0 = 4
\]
\[
\text{slope} = \frac{-2}{4} = -\frac{1}{2}
\]

4. **Between \( (4, 2) \) and \( (8, 0) \)**:

\[
\Delta y = 0 - 2 = -2
\]
\[
\Delta x = 8 - 4 = 4
\]
\[
\text{slope} = \frac{-2}{4} = -\frac{1}{2}
\]

5. **Between \( (8, 0) \) and \( (12, -2) \)**:

\[
\Delta y = -2 - 0 = -2
\]
\[
\Delta x = 12 - 8 = 4
\]
\[
\text{slope} = \frac{-2}{4} = -\frac{1}{2}
\]

### Conclusion

The slope (change in \( y \) over change in \( x \)) between all points is consistently \( -\frac{1}{2} \), indicating a linear relationship between \( x \) and \( y \).