A linear function has a constant rate of change, which means that the change in the value of y is proportional to the change in the value of x. This means that if we plot the points (x,y) for a linear function, they will fall on a straight line.
To determine which of the tables represents a linear function, we can calculate the rate of change between the points. If the rate of change is the same for all pairs of points, then the table represents a linear function.
For the first table:
Rate of change between (−2,4) and (−1,1) = (1−4)/(−1−(−2)) = −3
Rate of change between (−1,1) and (0,−2) = (−2−1)/(0−(−1)) = −3
Rate of change between (0,−2) and (1,−5) = (−5−(−2))/(1−0) = −3
Rate of change between (1,−5) and (2,−8) = (−8−(−5))/(2−1) = −3
Since the rate of change is constant (equal to −3) for all pairs of points, the first table represents a linear function.
For the second table:
Rate of change between (−2,4) and (−1,1) = (1−4)/(−1−(−2)) = −3
Rate of change between (−1,1) and (0,0) = (0−1)/(0−(−1)) = 1
Rate of change between (0,0) and (1,1) = (1−0)/(1−0) = 1
Rate of change between (1,1) and (2,4) = (4−1)/(2−1) = 3
Since the rate of change is not constant (it changes from −3 to 1 to 3), the second table does not represent a linear function.
For the third table:
Rate of change between (2,−2) and (2,−1) is undefined since x does not change.
Rate of change between (2,−1) and (0,0) = (0−(−1))/(0−2) = 0.5
Rate of change between (0,0) and (2,2) = (2−0)/(2−0) = 1
Rate of change between (2,2) and (2,−2) is undefined since x does not change.
Since the rate of change is not constant (it changes from undefined to 0.5 to 1 to undefined), the third table does not represent a linear function.
For the fourth table:
Rate of change between (0,−2) and (1,1) = (1−(−2))/(1−0) = 3
Rate of change between (1,1) and (2,0) = (0−1)/(2−1) = −1
Rate of change between (2,0) and (3,1) = (1−0)/(3−2) = 1
Rate of change between (3,1) and (4,−2) = (−2−1)/(4−3) = −3
Since the rate of change is not constant (it changes from 3 to −1 to 1 to −3), the fourth table does not represent a linear function.
Therefore, the only table that represents a linear function is the first one.