To determine the initial value and the rate of change of the linear function given the points (0.5, 1) and (-1.5, -3), we can follow these steps:
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Initial Value: The initial value of a linear function is typically the y-coordinate when x = 0. Since the point (0.5, 1) does not have x = 0, we would have to determine the y-intercept (where x = 0) using the points provided.
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Rate of Change (Slope): The slope (rate of change) of a line can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (-1.5, -3) and (0.5, 1):
\[ m = \frac{1 - (-3)}{0.5 - (-1.5)} = \frac{1 + 3}{0.5 + 1.5} = \frac{4}{2} = 2 \]
So, the rate of change is 2.
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Finding Initial Value (y-intercept): We can use the slope-intercept form of the line, which is:
\[ y = mx + b \]
where \( m = 2 \) (slope) and \( b \) is the y-intercept. To find \( b \), we can use one of the points. Let's use the point (0.5, 1):
\[ 1 = 2(0.5) + b \] \[ 1 = 1 + b \] \[ b = 1 - 1 = 0 \]
Hence, the initial value is 0.
Therefore, the final answers are:
- The initial value is (0),
- The rate of change is (2).