To find the initial value and the rate of change from the given points \((-3, 4)\), \((0, 3)\), \((3, 2)\), and \((6, 1)\), we can interpret these points as representing a linear function.
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Initial Value: This can be interpreted as the y-value when \(x=0\). From the point \((0, 3)\), we see that:
\[ \text{Initial Value} = 3 \]
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Rate of Change: The rate of change can be determined by calculating the slope of the line connecting the points. The slope (or rate of change) \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's calculate the slope between the points \((0, 3)\) and \((3, 2)\):
\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} \approx -\frac{1}{3} \]
We can verify this by checking the slope between other pairs, like \((3, 2)\) and \((6, 1)\):
\[ m = \frac{1 - 2}{6 - 3} = \frac{-1}{3} \]
The slope remains consistent.
Thus, we have:
- Initial Value: \(3\)
- Rate of Change: \(-\frac{1}{3}\)
Final answer:
- Initial Value is \((3)\)
- Rate of Change is \((-1/3)\)
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