the data in the table illustrate a linear function.
x | -3 | 0 | 3 | 6
y |-5 | -3 | -1 | 1
Yes, the data in the table does illustrate a linear function. We can see this by observing that for every increase in x by 3 units, y increases by 2 units (from -5 to -3, from -3 to -1, and from -1 to 1). This indicates a constant rate of change between x and y, which is a fundamental characteristic of linear functions.
To determine if the data in the table represents a linear function, we need to check if there is a constant rate of change between the x-values and the corresponding y-values.
In this case, let's calculate the rate of change by finding the difference in y-values divided by the difference in x-values.
Rate of change = (change in y) / (change in x)
For the first set of data points (-3, -5) and (0, -3):
(change in y) = -3 - (-5) = 2
(change in x) = 0 - (-3) = 3
Rate of change = 2 / 3
For the second set of data points (0, -3) and (3, -1):
(change in y) = -1 - (-3) = 2
(change in x) = 3 - 0 = 3
Rate of change = 2 / 3
For the third set of data points (3, -1) and (6, 1):
(change in y) = 1 - (-1) = 2
(change in x) = 6 - 3 = 3
Rate of change = 2 / 3
Since the rate of change is the same for all the data points, we can conclude that the data in the table represents a linear function.