To determine the rate of change (slope) and the initial value (y-intercept) of the linear function from the given points (2, 22) and (8, 25), we can use the following formulas:
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Rate of Change (Slope): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) = (2, 22) \) and \( (x_2, y_2) = (8, 25) \).
Plugging in the values: \[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
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Initial Value (y-intercept): To find the initial value (b), we can use the formula for the linear function in slope-intercept form: \[ y = mx + b \] We can substitute one of the points and the slope into this equation. Using the point (2, 22): \[ 22 = \frac{1}{2}(2) + b \] Simplifying: \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
Thus, the rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).
Final answer:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)