To determine the initial value and the rate of change of the linear function given the two points (2, 22) and (8, 25), we can follow these steps:
-
Identify Points: The points are \((x_1, y_1) = (2, 22)\) and \((x_2, y_2) = (8, 25)\).
-
Calculate the Rate of Change (Slope): The rate of change (slope) \(m\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values: \[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
-
Determine the Initial Value (y-intercept): The equation of a linear function can be written in the slope-intercept form: \[ y = mx + b \] We know \(m = \frac{1}{2}\). To find \(b\) (the initial value or y-intercept), we can substitute one of the points into the equation. Using the point (2, 22): \[ 22 = \frac{1}{2}(2) + b \] \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
Thus, the initial value (y-intercept) is \(21\) and the rate of change (slope) is \(\frac{1}{2}\).
Summary
- Initial value (y-intercept) \(b = 21\)
- Rate of change (slope) \(m = \frac{1}{2}\)