To find the initial value and the rate of change of a linear function given two points, we can use the two points you've provided: (-4, -10/3) and (-1, -4/3).
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Calculate the Rate of Change (Slope): The formula for the slope (rate of change) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1) = (-4, -10/3)\) and \((x_2, y_2) = (-1, -4/3)\).
Plugging in the values: \[ m = \frac{-4/3 - (-10/3)}{-1 - (-4)} = \frac{-4/3 + 10/3}{-1 + 4} = \frac{6/3}{3} = \frac{2}{3} \]
So, the rate of change is \(\frac{2}{3}\).
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Determine the Initial Value (y-intercept): We can use the slope-intercept form of a linear equation, \(y = mx + b\), where \(b\) is the y-intercept (initial value). We can use one of the points to find \(b\). Let's use the point \((-4, -10/3)\):
\[ -\frac{10}{3} = \frac{2}{3}(-4) + b \]
Which simplifies to: \[ -\frac{10}{3} = -\frac{8}{3} + b \]
Adding \(\frac{8}{3}\) to both sides gives: \[ b = -\frac{10}{3} + \frac{8}{3} = -\frac{2}{3} \]
So, the initial value is \(-\frac{2}{3}\) and the rate of change is \(\frac{2}{3}\).
Final Answer: The initial value is \(-\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).