To determine the initial value and the rate of change of the linear function given the coordinates of two points on the graph, we can use the following steps:
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Identify the points: The two points given are (-1.5, 3) and (0.5, 1).
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Find the rate of change (slope): The formula for slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (-1.5, 3) as \((x_1, y_1)\) and (0.5, 1) as \((x_2, y_2)\): \[ m = \frac{1 - 3}{0.5 - (-1.5)} = \frac{-2}{0.5 + 1.5} = \frac{-2}{2} = -1 \]
Thus, the rate of change (slope) is \( -1 \).
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Find the initial value (y-intercept): The y-intercept can be determined using the slope-intercept form of the line, \(y = mx + b\), where \(b\) is the y-intercept. We can use one of the points to find \(b\). Let’s use (0.5, 1): \[ 1 = -1(0.5) + b \] \[ 1 = -0.5 + b \implies b = 1 + 0.5 = 1.5 \]
So, based on this calculation, the initial value (y-intercept) is \(1.5\), and the rate of change (slope) is \(-1\).
However, the options you provided do not match with this calculation. If the choices you provided were related to a different situation or point set, please provide those details or the correct points for further assistance!