To determine the initial value and the rate of change of the linear function represented by the given points on the coordinate plane, we first need to find the slope (rate of change) and the y-intercept (initial value).
Step 1: Find the Rate of Change (Slope)
We can use the two points provided:
- \( P_1 \left( -4, -\frac{10}{3} \right) \)
- \( P_2 \left( -1, -\frac{4}{3} \right) \)
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting our points:
\[ m = \frac{-\frac{4}{3} - \left(-\frac{10}{3}\right)}{-1 - (-4)} \]
Calculating the difference in the y-coordinates:
\[ -\frac{4}{3} + \frac{10}{3} = \frac{6}{3} = 2 \]
Calculating the difference in the x-coordinates:
\[ -1 + 4 = 3 \]
Now substituting back to find the slope:
\[ m = \frac{2}{3} \]
Step 2: Find the Initial Value (y-intercept)
We can use the slope-intercept form of the equation of a line, \( y = mx + b \), where \( b \) is the y-intercept. We can plug in one of the points and the slope:
Using point \( P_1 \left( -4, -\frac{10}{3} \right) \):
\[ -\frac{10}{3} = \left( \frac{2}{3} \right)(-4) + b \]
Calculating:
\[ -\frac{10}{3} = -\frac{8}{3} + b \]
Adding \( \frac{8}{3} \) to both sides:
\[ b = -\frac{10}{3} + \frac{8}{3} = -\frac{2}{3} \]
Conclusion
Now we have:
- The initial value (y-intercept) \( b = -\frac{2}{3} \)
- The rate of change (slope) \( m = \frac{2}{3} \)
Thus, the answer is:
The initial value is negative \(\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).
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