Which table represents a linear function?
X 1 2 3 4 y -2 -6 -2 -6
The Correct Answer and Explanation is :
To determine whether a table represents a linear function, we must check if the relationship between the variables (x) and (y) produces a constant rate of change. A linear function is defined by the equation (y = mx + b), where (m) is the slope (rate of change) and (b) is the y-intercept.
Given the table:
[
\begin{array}{|c|c|}
\hline
x & y \
\hline
1 & -2 \
2 & -6 \
3 & -2 \
4 & -6 \
\hline
\end{array}
]
We can analyze the (y) values corresponding to each (x) value. To check for a constant rate of change, we can compute the differences in (y) for the changes in (x):
- For (x = 1) to (x = 2):
- Change in (y) = (-6 – (-2) = -6 + 2 = -4)
- For (x = 2) to (x = 3):
- Change in (y) = (-2 – (-6) = -2 + 6 = 4)
- For (x = 3) to (x = 4):
- Change in (y) = (-6 – (-2) = -6 + 2 = -4)
Now, we look at the changes:
- From (1) to (2): (-4)
- From (2) to (3): (4)
- From (3) to (4): (-4)
The differences are (-4), (4), and (-4). Since these changes are not consistent (i.e., they do not form a constant value), the relationship between (x) and (y) in this table does not represent a linear function.
For a function to be linear, the changes in (y) (output) must be proportional to the changes in (x) (input). In this case, the alternating values of (y) show that it does not exhibit a steady increase or decrease as (x) changes, thus indicating that this relationship does not define a linear function.