To determine the initial value (y-intercept) and the rate of change (slope) of the linear function represented by the line on the coordinate plane, we can follow these steps:
- Calculate the Rate of Change (Slope):
The slope (m) of the line can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
From the points provided:
- Point 1: \((-4, -\frac{10}{3})\)
- Point 2: \((-1, -\frac{4}{3})\)
Using these points:
- \(y_1 = -\frac{10}{3}, y_2 = -\frac{4}{3}\)
- \(x_1 = -4, x_2 = -1\)
Now substituting into the slope formula: \[ m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{3} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]
Thus, the rate of change (slope) is \(\frac{2}{3}\).
- Determine the Initial Value (y-intercept): The initial value (y-intercept) can be found using the slope-intercept form of a linear equation: \[ y = mx + b \] Where \(b\) is the y-intercept. We can rearrange the formula using one of the points to find \(b\).
Using point \((-1, -\frac{4}{3})\): \[ -\frac{4}{3} = \frac{2}{3}(-1) + b \] \[ -\frac{4}{3} = -\frac{2}{3} + b \] \[ b = -\frac{4}{3} + \frac{2}{3} = -\frac{4 - 2}{3} = -\frac{2}{3} \]
So, the initial value (y-intercept) is \(-\frac{2}{3}\).
- Conclusion: The initial value is \(-\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).
From the options provided: The correct response is: The initial value is negative \(\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).