- Analyze the provided table to identify coordinate pairs of books sold and profit.
- Use the slope formula \(m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\) to determine the constant rate of change between pairs of points.
- Verify that the rate of change is uniform across all given data points to confirm the function is linear.
- Conclude that the rate of change is \(\boxed{\$1.50}\) per book.
Explanation
- Analyzing the Provided Data
The objective is to find the rate of change of the profit function \(f(x)\), which represents the profit based on the number of books sold \(x\). The data provided in the table consists of several coordinate pairs \((x, f(x))\): (100, 50.00), (250, 275.00), (300, 350.00), and (350, 425.00). The rate of change of a linear function is constant and represents how much the output changes for each unit increase in input, defined by the formula:
\[m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
- Calculating the Rate of Change
To verify that the function is linear, we calculate the rate of change between consecutive points from the table:
- Between (100, 50.00) and (250, 275.00):
\[m_1 = \frac{275 - 50}{250 - 100} = \frac{225}{150} = 1.5\]
- Between (250, 275.00) and (300, 350.00):
\[m_2 = \frac{350 - 275}{300 - 250} = \frac{75}{50} = 1.5\]
- Between (300, 350.00) and (350, 425.00):
\[m_3 = \frac{425 - 350}{350 - 300} = \frac{75}{50} = 1.5\]
The rate of change is constant at \(\$1.50\) per book.
- Conclusion
The calculated rate of change for the function is \(\$1.50\) per book. Comparing this to the given options, we identify that the correct rate of change is \(\$1.50\) per book. Therefore, the final answer is \(\boxed{\$1.50}\) per book.
Examples
In real-life scenarios, identifying the rate of change is essential for understanding relationships between variables, such as calculating the cost per unit in a business or predicting future values based on past performance. For instance, if you run a small business selling handmade crafts, determining the profit per unit helps you estimate how much profit you will earn based on the volume of items sold, allowing for better financial planning and growth projections. This mathematical approach helps you make informed decisions based on observed data.