The average rate of change produces the same value for any interval of the domain for linear functions.
Explanation:
A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The average rate of change between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is calculated as:
\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
For a linear function:
\[ f(x_1) = mx_1 + b \quad \text{and} \quad f(x_2) = mx_2 + b \]
Substituting these into the average rate of change formula gives:
\[ \text{Average Rate of Change} = \frac{(mx_2 + b) - (mx_1 + b)}{x_2 - x_1} = \frac{mx_2 - mx_1}{x_2 - x_1} = \frac{m(x_2 - x_1)}{x_2 - x_1} = m \]
Since \( m \) (the slope) is constant, the average rate of change remains the same for any interval in the domain of the function. This consistency occurs because linear functions have a constant slope, leading to a uniform change in the function value relative to changes in the input.