To determine the initial value and the rate of change of the line, we can analyze the points that the line passes through and calculate the slope (rate of change) of the function.
The two points provided are:
- \((-4, -\frac{10}{3})\)
- \((-1, -\frac{4}{3})\)
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 \( (x_1, y_1) = (-4, -\frac{10}{3}) \) and \( (x_2, y_2) = (-1, -\frac{4}{3}) \). Plugging in these values:
\[ m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]
So, the rate of change (slope) is \( \frac{2}{3} \).
Step 2: Determine the initial value (y-intercept)
To find the initial value (y-intercept), we can use the slope-intercept form of the equation \( y = mx + b \). We can substitute one of the points and the slope into this equation to find \( b \):
Using point \((-1, -\frac{4}{3})\):
\[ -\frac{4}{3} = \frac{2}{3}(-1) + b \] \[ -\frac{4}{3} = -\frac{2}{3} + b \] \[ b = -\frac{4}{3} + \frac{2}{3} = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3} \]
Thus, the initial value is \( -\frac{2}{3} \).
Putting it all together:
The initial value is \( -\frac{2}{3} \), and the rate of change is \( \frac{2}{3} \).
The correct response is: The initial value is negative \(\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).