Question

(-3,4) (0,3) (3,2) (6,1)


the initial value is (), and the rate of change is ()

Answer 1

To find the initial value and the rate of change from the given points \((-3, 4)\), \((0, 3)\), \((3, 2)\), and \((6, 1)\), we can interpret these points as representing a linear function.

1. **Initial Value:** This can be interpreted as the y-value when \(x=0\). From the point \((0, 3)\), we see that:

\[
\text{Initial Value} = 3
\]

2. **Rate of Change:** The rate of change can be determined by calculating the slope of the line connecting the points. The slope (or rate of change) \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:

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

Let's calculate the slope between the points \((0, 3)\) and \((3, 2)\):

\[
m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \approx -\frac{1}{3}
\]

We can verify this by checking the slope between other pairs, like \((3, 2)\) and \((6, 1)\):

\[
m = \frac{1 - 2}{6 - 3} = \frac{-1}{3}
\]

The slope remains consistent.

Thus, we have:

- **Initial Value:** \(3\)
- **Rate of Change:** \(-\frac{1}{3}\)

Final answer:

- Initial Value is \((3)\)
- Rate of Change is \((-1/3)\)