To determine the initial value and rate of change of the linear function based on the provided points, we can follow these steps:
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Identify Points: The points given are:
- A: (-3, 4)
- B: (0, 3)
- C: (3, 2)
- D: (6, 1)
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Calculate the Slope (Rate of Change): The slope (m) of a linear function is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We can use any two points. Let's use the first two points, A and B:
\[ m = \frac{3 - 4}{0 - (-3)} = \frac{-1}{3} \approx -0.333 \]
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Confirm the Slope with Other Points: We can check the slope between other pairs to ensure consistency:
Between B (0, 3) and C (3, 2): \[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \approx -0.333 \]
Between C (3, 2) and D (6, 1): \[ m = \frac{1 - 2}{6 - 3} = \frac{-1}{3} \approx -0.333 \]
So, the rate of change (slope) is consistently \(-0.333\).
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Find the Initial Value: The initial value (y-intercept) can be identified from the point where the line crosses the y-axis. From point B (0, 3), we see that when \(x = 0\), \(y = 3\).
So, the initial value is:
- Initial Value (y-intercept) = 3.
Final Results
- Initial Value: \(3.000\)
- Rate of Change (Slope): \(-0.333\)
Thus, the linear function can be described as: \[ y = -0.333x + 3 \]
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