To determine the initial value and rate of change of the linear function represented by the given points (-3, 4), (0, 3), (3, 2), and (6, 1), we can use two of the points to calculate the slope (rate of change) and then find the linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept (initial value).
Step 1: Calculate the rate of change (slope)
The slope \( m \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's choose the points (0, 3) and (3, 2):
- \( (x_1, y_1) = (0, 3) \)
- \( (x_2, y_2) = (3, 2) \)
Now plug in the values:
\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \]
So the rate of change (slope) is \( -\frac{1}{3} \).
Step 2: Find the initial value (y-intercept)
The y-intercept \( b \) is the value of \( y \) when \( x = 0 \). From the point (0, 3), we can see that:
\[ b = 3 \]
Conclusion
The initial value (y-intercept) is \( 3 \), and the rate of change (slope) is \( -\frac{1}{3} \).
Therefore, the linear function can be expressed as:
\[ y = -\frac{1}{3}x + 3 \]