- Analyze the given table of values and calculate the differences between consecutive revenue amounts to confirm a constant rate of change.
- Use the slope-intercept form \(f(n) = mn + b\) and one of the given points to calculate the slope \(m = -0.2\) and y-intercept \(b = 2.8\).
- Compare the derived equation \(f(n) = -0.2n + 2.8\) with the provided options to identify the correct function (b).
- Construct the graph for Part B by plotting the four given points on a coordinate plane and drawing the line representing the function.
- The final function is \(\boxed{f(n) = -0.2n + 2.8}\).
Explanation
- Analyzing the data for a linear relationship
The objective is to identify the correct linear function \(f(n)\) from the given options that models the gross revenue based on the provided table. The table provides data for weeks \(n = 1, 2, 3, 4\) with corresponding revenues \(f(n) = 2.6, 2.4, 2.2, 2.0\). We analyze the change in revenue over time to determine the nature of the relationship. Calculating the differences:
\(f(2) - f(1) = 2.4 - 2.6 = -0.2\)
\(f(3) - f(2) = 2.2 - 2.4 = -0.2\)
\(f(4) - f(3) = 2.0 - 2.2 = -0.2\)
Since the difference between consecutive revenue values is constant, the relationship is linear.
- Determining the linear function
To find the linear function \(f(n) = mn + b\), we need the slope \(m\) and the y-intercept \(b\).
The slope \(m\) is the constant difference between consecutive terms, which we already found to be \(-0.2\).
Now, we use one of the data points, for example, \((n, f(n)) = (1, 2.6)\), to solve for \(b\):
\(2.6 = -0.2(1) + b\)
\(2.6 = -0.2 + b\)
\(b = 2.6 + 0.2 = 2.8\)
Thus, the linear function is \(f(n) = -0.2n + 2.8\). Comparing this to the given options:
A\(f(n) = -2n - 2.8\)
B\(f(n) = -0.2n + 2.8\)
C\(f(n) = 0.3n + 2\)
D\(f(n) = -0.2n^2 - 2\)
Option (b) is the correct function.
- Graphing the function
To graph the function \(f(n) = -0.2n + 2.8\), we plot the four points \((1, 2.6)\), \((2, 2.4)\), \((3, 2.2)\), and \((4, 2.0)\) on a coordinate plane where the horizontal axis represents 'Week' (\(n\)) and the vertical axis represents 'Gross revenue' (\(f(n)\)). Since the function is linear, these points lie on a straight line with a negative slope, indicating a consistent decrease in revenue over the four weeks. The line starts at \((0, 2.8)\) and moves downward as \(n\) increases.
- Conclusion
The correct function representing the sequence of gross revenue is \(f(n) = -0.2n + 2.8\). The relationship is confirmed as linear, with a constant rate of change of \(-0.2\) million dollars per week. The final answer for Part A is option (b), and the graph in Part B is formed by a line passing through the points \((1, 2.6), (2, 2.4), (3, 2.2), (4, 2.0)\). \(\boxed{f(n) = -0.2n + 2.8}\).
Examples
Understanding linear relationships is essential in business for forecasting. For example, if a store notices that its daily sales of a specific item are decreasing at a steady rate due to a seasonal change, the store manager can model this trend using a linear equation. By plotting the daily sales, the manager can predict when the stock might run out or when they should stop reordering the item, allowing for better inventory management and financial planning.