How to Calculate and Interpret Average Rate of Change from a Machine Value Graph

Question

Find and interpret the average rate of change illustrated in the graph. Select the correct choice below and fill in the answer box to complete your choice. A. The average rate of change is $____ per year. The machine value is increasing $____ each year. B. The average rate of change is $____ per year. The machine value is decreasing $____ each year. The graph has a y-axis labeled 'Value of Machine (in thousands of dollars)' ranging from 0 to 20, and an x-axis labeled 'Year' ranging from 0 to 20. The line passes through the points (0, 16) and (4, 0).

Accepted Answer

Option B; The average rate of change is $-4000 per year. The machine value is decreasing $4000 each year.

Answer

Step 1: Identify two clear points on the graph. The line passes through (0, 16) and (4, 0). Note that the y-axis is in thousands of dollars, so (0, 16) means $16,000 at year 0, and (4, 0) means $0 at year 4. Step 2: Use the average rate of change formula: $\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1) = (0, 16000)$ and $(x_2, y_2) = (4, 0)$. Step 3: Substitute the values into the formula: $\frac{0 - 16000}{4 - 0} = \frac{-16000}{4} = -4000$. Step 4: Interpret the result. The negative sign indicates a decrease, so the machine's value decreases by $4000 each year, which corresponds to option B.