To determine the initial value and the rate of change of the linear function represented by the line in the coordinate plane, we will follow these steps:
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Calculate the slope (rate of change) using the two points provided: \((-4, -\frac{10}{3})\) and \((-1, -\frac{4}{3})\).
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \[ x_1 = -4, \quad y_1 = -\frac{10}{3}, \quad x_2 = -1, \quad y_2 = -\frac{4}{3} \] Substituting these values into the slope formula gives us: \[ m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]
Therefore, the rate of change is \(\frac{2}{3}\).
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Find the initial value. The initial value is the y-intercept of the line (the value of \(y\) when \(x = 0\)). We can use the equation of the line in slope-intercept form, \(y = mx + b\), or we can substitute one of the points to solve for \(b\).
Using the slope \(m = \frac{2}{3}\) and the point \((-1, -\frac{4}{3})\): \[ -\frac{4}{3} = \frac{2}{3}(-1) + b \] \[ -\frac{4}{3} = -\frac{2}{3} + b \] \[ b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3} \]
Thus, the initial value (y-intercept) is \(-\frac{2}{3}\).
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Summarizing the results:
- The initial value is \(-\frac{2}{3}\).
- The rate of change is \(\frac{2}{3}\).
Given this analysis, the correct response choice based on the options provided is:
The initial value is \( -2 \), and the rate of change is \( \frac{2}{3} \).
1 answer