- A linear function has a constant rate of change (\(\frac{\Delta y}{\Delta x}\)) between consecutive points.
- By calculating the differences between successive \(y\)-values for each table, we observe that the first table has alternating differences, and the second table has changing differences (\(-3, -4\)).
- The third table exhibits a constant difference of \(-8\) for each step of \(x\), confirming that it represents a linear function.
- The third table is the correct one.
- \(\boxed{\text{Table 3}}\)
Explanation
- Understanding Linear Functions
A linear function is a relationship between two variables, \(x\) and \(y\), that can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. A fundamental property of a linear function is that for equal intervals of \(x\), the corresponding changes in \(y\) (denoted as \(\Delta y\)) must be constant. In our provided tables, the increment in \(x\) is always constant at \(\Delta x = 1\). Therefore, we need to check which table exhibits a constant difference \(\Delta y = y_{i+1} - y_i\) between consecutive values.
- Analyzing the First Table
Let's evaluate the differences in \(y\) for the first table:
- Between \(x=1\) and \(x=2\): \(\Delta y = -6 - (-2) = -4\)
- Between \(x=2\) and \(x=3\): \(\Delta y = -2 - (-6) = 4\)
Since the differences are not the same (\(-4
eq 4\)), this table does not represent a linear function.
- Analyzing the Second Table
Now, evaluate the differences in \(y\) for the second table:
- Between \(x=1\) and \(x=2\): \(\Delta y = -5 - (-2) = -3\)
- Between \(x=2\) and \(x=3\): \(\Delta y = -9 - (-5) = -4\)
Since the differences are not the same (\(-3
eq -4\)), this table does not represent a linear function.
- Analyzing the Third Table
Finally, evaluate the differences in \(y\) for the third table:
- Between \(x=1\) and \(x=2\): \(\Delta y = -10 - (-2) = -8\)
- Between \(x=2\) and \(x=3\): \(\Delta y = -18 - (-10) = -8\)
- Between \(x=3\) and \(x=4\): \(\Delta y = -26 - (-18) = -8\)
Since the difference \(\Delta y = -8\) is constant for all intervals, the relationship defined in the third table is a linear function with a slope of \(-8\).
- Conclusion
The third table satisfies the condition for a linear function, as it maintains a constant rate of change. We can identify it as the correct table representing the linear function. The final answer is the third table.
Examples
Linear functions are used in real-life to represent constant rates of change. For example, if you are filling a pool with water at a constant rate, the volume of water in the pool over time will follow a linear relationship. By knowing the volume at different time intervals, you can create a table and check for a constant difference in volume to confirm that the filling rate is indeed constant. This allows you to predict the time it will take to fill the pool completely based on the slope of the linear function.