Marco is driving to the Grand Canyon. His distance from the Grand Canyon decreases 150 mi every 3 h . After 4 h , his distance from the Grand Canyon is \( \mathbf{2 0 0 ~ m i} \). Marco's distance from the Grand Canyon in miles, \( y \), is a function of the number of hours he drives, \( x \). What is the rate of change? Find the change in Marco's distance each hour. rate of change: \( \square \) \( -50 \) What is the initial value? Find Marco's distance from the Grand Canyon when he starts to drive. initial value: 400 Write an equation to represent the function. \[ y=1 x+\square \]

To solve the problem, we need to extract the known conditions and then find the required values step by step. 1. **Known Conditions:** - Marco decreases his distance from the Grand Canyon by 150 miles every 3 hours. - After 4 hours, his distance from the Grand Canyon is 200 miles. 2. **Rate of Change:** - The rate of change is the distance decreased per hour. Since Marco decreases his distance by 150 miles in 3 hours, we can calculate the rate of change as follows: \[ \text{Rate of Change} = \frac{-150 \text{ miles}}{3 \text{ hours}} = -50 \text{ miles per hour} \] Thus, the rate of change is \(-50\). 3. **Initial Value:** - We know that after 4 hours, Marco's distance from the Grand Canyon is 200 miles. To find the initial distance when he starts driving (at \(x = 0\)), we can use the information that he decreases his distance by 50 miles each hour. - After 4 hours, the total decrease in distance is: \[ \text{Total Decrease} = 4 \text{ hours} \times 50 \text{ miles/hour} = 200 \text{ miles} \] - Therefore, the initial distance \(y_0\) when he starts driving is: \[ y_0 = 200 \text{ miles} + 200 \text{ miles} = 400 \text{ miles} \] Thus, the initial value is \(400\). 4. **Equation to Represent the Function:** - The function \(y\) can be expressed in the slope-intercept form \(y = mx + b\), where \(m\) is the rate of change and \(b\) is the initial value. - Here, \(m = -50\) and \(b = 400\). Therefore, the equation is: \[ y = -50x + 400 \] Now, we can summarize the answers: - Rate of change: \(-50\) - Initial value: \(400\) - Equation: \[ y = -50x + 400 \]