- Define a linear function as one with a constant rate of change.
- Calculate the slopes between each consecutive pair of points from the table, finding each to be equal to \(-2\).
- Conclude that the function is linear because it exhibits a constant rate of change.
- The correct choice is: \(\boxed{\text{No, because it has a constant rate of change.}}\)
Explanation
- Analyzing the definition of a linear function
To determine if a function is non-linear, we must check if its rate of change (or slope) is constant between consecutive points in a table. A linear function possesses a constant rate of change, while a non-linear function does not. We are given the following points \((x, y)\): \((6, 4)\), \((7, 2)\), \((8, 0)\), and \((9, -2)\). The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
- Calculating the rate of change
We calculate the slope between each consecutive pair of points provided in the table:
- Between \((6, 4)\) and \((7, 2)\):
\[m_1 = \frac{2 - 4}{7 - 6} = \frac{-2}{1} = -2\]
- Between \((7, 2)\) and \((8, 0)\):
\[m_2 = \frac{0 - 2}{8 - 7} = \frac{-2}{1} = -2\]
- Between \((8, 0)\) and \((9, -2)\):
\[m_3 = \frac{-2 - 0}{9 - 8} = \frac{-2}{1} = -2\]
- Conclusion and Final Answer
Comparing the calculated slopes, we observe that \(m_1 = m_2 = m_3 = -2\). Since the rate of change is constant across all consecutive points, the function is linear. Therefore, the function is not non-linear. The correct answer is: "No, because it has a constant rate of change."
Examples
Understanding linear relationships is essential in everyday life, such as when calculating the cost of a rental service that charges a fixed fee per unit of time or distance. For example, if a taxi charges a base fare plus a constant rate per kilometer, the relationship between the distance traveled and the total cost is a linear function. By recognizing a constant rate of change in such data, we can accurately predict future costs or determine the underlying pricing structure of various services.