Let \( f(x)-2 x-2 \). (a) Find the average rate of change from 8 to 9 . (b) Find an equation of the secant line containing \( (8, f(8)) \) and \( (9, f(9) \) ). (a) The average rate of change from 8 to 9 is \( \square \). (Simplify your answer.)

To find the average rate of change of the function \( f(x) = 2x - 2 \) from \( x = 8 \) to \( x = 9 \), we calculate: \[ f(8) = 2(8) - 2 = 16 - 2 = 14 \] \[ f(9) = 2(9) - 2 = 18 - 2 = 16 \] The average rate of change is given by: \[ \text{Average rate of change} = \frac{f(9) - f(8)}{9 - 8} = \frac{16 - 14}{1} = 2 \] So the average rate of change from 8 to 9 is \( \boxed{2} \). Next, to find the equation of the secant line that passes through the points \( (8, f(8)) = (8, 14) \) and \( (9, f(9)) = (9, 16) \), we use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \] Where \( m \) is the slope. Since we calculated the average rate of change (slope) to be \( 2 \): Plugging in one of the points, we can use \( (8, 14) \): \[ y - 14 = 2(x - 8) \] Expanding this, we get: \[ y - 14 = 2x - 16 \] \[ y = 2x - 2 \] Thus, the equation of the secant line is \( \boxed{y = 2x - 2} \).