To determine the initial value and the rate of change of the linear function defined by the points (6,1), (3,3), and (0,5), we can follow these steps:
-
Identify the initial value (y-intercept): The y-intercept of a linear function occurs where \(x = 0\). From the given points, the point (0,5) indicates that the initial value (y-intercept) is 5.
-
Determine the rate of change (slope): The slope (rate of change) of a linear function can be calculated using any two points on the line. The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use the points (6,1) and (3,3) for this calculation:
Let \((x_1, y_1) = (6,1)\) and \((x_2, y_2) = (3,3)\). \[ m = \frac{3 - 1}{3 - 6} = \frac{2}{-3} = -\frac{2}{3} \]
Thus, the rate of change (slope) is \(-\frac{2}{3}\).
To summarize:
- The initial value (y-intercept) is 5.
- The rate of change (slope) is \(-\frac{2}{3}\).