- A linear function requires a constant rate of change between all consecutive points.
- Calculate the slopes: \(m_1 = 7\), \(m_2 = 5\), and \(m_3 = 6\).
- Since the slopes are different, the rate of change is not constant.
- The function is not linear: \(\boxed{\text{No, because it does not have a constant rate of change.}}\)
Explanation
- Analyzing the Linearity Requirement
A linear function \(y = mx + b\) is characterized by a constant rate of change, or slope, denoted by \(m\). Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
If the function represented by a table is linear, the slope calculated between any two consecutive points must be identical. To determine if our given function is linear, we will calculate the rate of change between consecutive points from the table.
- Calculating Slopes Between Consecutive Points
We evaluate the rate of change for each pair of consecutive points:
- Between \((10, -6)\) and \((11, 1)\):
\[m_1 = \frac{1 - (-6)}{11 - 10} = \frac{7}{1} = 7\]
- Between \((11, 1)\) and \((12, 6)\):
\[m_2 = \frac{6 - 1}{12 - 11} = \frac{5}{1} = 5\]
- Between \((12, 6)\) and \((13, 12)\):
\[m_3 = \frac{12 - 6}{13 - 12} = \frac{6}{1} = 6\]
- Determining Linearity and Selecting the Final Option
Comparing our results, we find that \(m_1 = 7\), \(m_2 = 5\), and \(m_3 = 6\). Since the rate of change is not constant (\(7
eq 5
eq 6\)), the function does not have a constant rate of change. Therefore, the function cannot be linear. Looking at the given options, the correct statement is: No, because it does not have a constant rate of change.
Examples
Understanding whether a function is linear helps in predicting future values based on a constant trend. For instance, if you are saving money at a constant rate every week, your total savings form a linear function over time. However, if the amount you save changes unpredictably each week, your savings pattern is non-linear, making it harder to estimate your future balance. Recognizing these patterns is essential in financial planning, data science, and everyday project management.