Question

find the rate of change
x
-2,-1,0,1,2

y
-3,-1,1,3,5

Answer 1

To find the rate of change of the values of \( y \) with respect to \( x \), we can use the formula for the average rate of change between two points on the graph, which is given by:

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

### Given Values:
- For \( x = -2, y = -3 \)
- For \( x = -1, y = -1 \)
- For \( x = 0, y = 1 \)
- For \( x = 1, y = 3 \)
- For \( x = 2, y = 5 \)

### Calculate Rate of Change:
1. Between \( (-2, -3) \) and \( (-1, -1) \):
\[
\text{Rate} = \frac{-1 - (-3)}{-1 - (-2)} = \frac{-1 + 3}{-1 + 2} = \frac{2}{1} = 2
\]

2. Between \( (-1, -1) \) and \( (0, 1) \):
\[
\text{Rate} = \frac{1 - (-1)}{0 - (-1)} = \frac{1 + 1}{0 + 1} = \frac{2}{1} = 2
\]

3. Between \( (0, 1) \) and \( (1, 3) \):
\[
\text{Rate} = \frac{3 - 1}{1 - 0} = \frac{2}{1} = 2
\]

4. Between \( (1, 3) \) and \( (2, 5) \):
\[
\text{Rate} = \frac{5 - 3}{2 - 1} = \frac{2}{1} = 2
\]

### Conclusion:
The rate of change of \( y \) with respect to \( x \) is constant and equal to \( 2 \) across all defined intervals. Thus, the function represented by these points is linear with a slope of \( 2 \).