To determine the Rate of Change and the Initial Value from the given table, we can analyze the values of \(x\) and \(y\).
The data points are:
\[ \begin{align*} x = 0, & \quad y = 50 \ x = 1, & \quad y = 250 \ x = 2, & \quad y = 450 \ x = 3, & \quad y = 650 \ \end{align*} \]
Initial Value: The initial value of \(y\) when \(x = 0\) is \(y = 50\).
Rate of Change: The rate of change (slope) can be calculated using points on the table. The rate of change can be calculated as the change in \(y\) divided by the change in \(x\) between two points.
Let's calculate the rate of change using the first two points:
- From \(x = 0\) to \(x = 1\): \[ \text{Rate of Change} = \frac{y(1) - y(0)}{x(1) - x(0)} = \frac{250 - 50}{1 - 0} = \frac{200}{1} = 200 \]
We can validate this rate of change for other intervals as well:
-
From \(x = 1\) to \(x = 2\): \[ \frac{450 - 250}{2 - 1} = \frac{200}{1} = 200 \]
-
From \(x = 2\) to \(x = 3\): \[ \frac{650 - 450}{3 - 2} = \frac{200}{1} = 200 \]
In all cases, the rate of change is consistently \(200\).
Thus, we conclude:
- Initial Value: 50
- Rate of Change: 200
Therefore, the correct choice is:
Initial Value: 50, Rate of Change: 200