To determine the initial value and the rate of change of the linear function described in the problem, we’ll do the following:
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Find the slope (rate of change) using the two points given:
- Point 1: \((-4, -\frac{10}{3})\)
- Point 2: \((-1, -\frac{4}{3})\)
The formula for 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 the values: \[ m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]
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Determine the initial value (y-intercept) using the slope-intercept form \(y = mx + b\): We have calculated \(m = \frac{2}{3}\). We can use one of the points to solve for \(b\).
Let's use the point \((-1, -\frac{4}{3})\): \[ -\frac{4}{3} = \frac{2}{3}(-1) + b \] \[ -\frac{4}{3} = -\frac{2}{3} + b \] Adding \(\frac{2}{3}\) to both sides: \[ b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3} \]
Thus, the initial value (y-intercept) is \(-\frac{2}{3}\) and the rate of change (slope) is \(\frac{2}{3}\).
The correct response is:
The initial value is negative \(\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).