Question

determine inital value and rate of change of linear function, round to the nearest three decimal places as needed
-3,4
0,3
3,2
6,1

Answer 1

To determine the initial value and rate of change of the linear function based on the provided points, we can follow these steps:

1. **Identify Points:** The points given are:
- A: (-3, 4)
- B: (0, 3)
- C: (3, 2)
- D: (6, 1)

2. **Calculate the Slope (Rate of Change):** The slope (m) of a linear function is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can use any two points. Let's use the first two points, A and B:

\[
m = \frac{3 - 4}{0 - (-3)} = \frac{-1}{3} \approx -0.333
\]

3. **Confirm the Slope with Other Points:** We can check the slope between other pairs to ensure consistency:

Between B (0, 3) and C (3, 2):
\[
m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \approx -0.333
\]

Between C (3, 2) and D (6, 1):
\[
m = \frac{1 - 2}{6 - 3} = \frac{-1}{3} \approx -0.333
\]

So, the rate of change (slope) is consistently \(-0.333\).

4. **Find the Initial Value:** The initial value (y-intercept) can be identified from the point where the line crosses the y-axis. From point B (0, 3), we see that when \(x = 0\), \(y = 3\).

So, the initial value is:
- **Initial Value (y-intercept) = 3.**

### Final Results
- **Initial Value:** \(3.000\)
- **Rate of Change (Slope):** \(-0.333\)

Thus, the linear function can be described as:
\[
y = -0.333x + 3
\]