To determine the rate of change (\(m\)) and the initial value (\(b\)) of the linear function from the given table, we can follow these steps.
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Calculate the Rate of Change (Slope \(m\)): The formula for the rate of change (slope) 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 \((2, 22)\) and \((8, 25)\):
- \(x_1 = 2\), \(y_1 = 22\)
- \(x_2 = 8\), \(y_2 = 25\)
Substitute these values into the formula:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
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Determine the Initial Value (\(b\)): The initial value (\(b\)) is the value of \(y\) when \(x = 0\). We can use the slope-intercept form of the equation of a line, which is \(y = mx + b\).
We can use one of the points to find \(b\). Let's use the point \((2, 22)\):
\[ 22 = \left(\frac{1}{2}\right)(2) + b \]
Simplifying this:
\[ 22 = 1 + b \]
Therefore, \(b = 22 - 1 = 21\).
So, from the calculations:
- The rate of change is \(m = \frac{1}{2}\).
- The initial value is \(b = 21\).
Thus, the final answer is:
- The rate of change is \(m = \frac{1}{2}\), and the initial value is \(b = 21\).