Question

find the rate of change
x
-2,0,2,6,10

y
-3,3,9,21,33

Answer 1

To find the rate of change between the given points, we can use the formula for the slope (rate of change) between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Let's calculate the rate of change for each pair of consecutive points.

### Given Data Points:
\[
\begin{align*}
( -2, -3), \\
( 0, 3), \\
( 2, 9), \\
( 6, 21), \\
(10, 33)
\end{align*}
\]

### Rate of Change Calculations:
1. Between \((-2, -3)\) and \((0, 3)\):
\[
\text{slope} = \frac{3 - (-3)}{0 - (-2)} = \frac{3 + 3}{0 + 2} = \frac{6}{2} = 3
\]

2. Between \((0, 3)\) and \((2, 9)\):
\[
\text{slope} = \frac{9 - 3}{2 - 0} = \frac{6}{2} = 3
\]

3. Between \((2, 9)\) and \((6, 21)\):
\[
\text{slope} = \frac{21 - 9}{6 - 2} = \frac{12}{4} = 3
\]

4. Between \((6, 21)\) and \((10, 33)\):
\[
\text{slope} = \frac{33 - 21}{10 - 6} = \frac{12}{4} = 3
\]

### Summary of Rates of Change:

- Rate of change from \((-2, -3)\) to \((0, 3)\): **3**
- Rate of change from \((0, 3)\) to \((2, 9)\): **3**
- Rate of change from \((2, 9)\) to \((6, 21)\): **3**
- Rate of change from \((6, 21)\) to \((10, 33)\): **3**

The rate of change between each pair of consecutive points is constant at **3**.