The table shows a function. Is the function linear or nonlinear?
x y
–4 –3
–1/2 2
3 3
To determine if a function is linear or nonlinear, we need to check if the rate of change between the x and y values is constant.
In this case, let's calculate the rate of change between the x and y values:
Rate of change = (change in y) / (change in x)
For -4 to -1/2:
(change in y) = 2 - (-3) = 5
(change in x) = -1/2 - (-4) = -1/2 + 4 = 7/2
Rate of change = 5 / (7/2) = 10/7
For -1/2 to 3:
(change in y) = 3 - 2 = 1
(change in x) = 3 - (-1/2) = 3 + 1/2 = 7/2
Rate of change = 1 / (7/2) = 2/7
Since the rate of change between the x and y values is not constant (10/7 and 2/7 are not equal), the function is nonlinear.
To determine whether the function is linear or nonlinear, we can observe the relationship between the x-values and the corresponding y-values in the table.
In a linear function, the relationship between x and y is consistent and can be expressed using a straight line. This means that we should see a constant difference or ratio between the x and y-values.
Let's calculate the differences between the y-values for each pair of x-values:
For (-4, -3) and (-1/2, 2):
The difference in y-values is 2 - (-3) = 5.
For (-1/2, 2) and (3, 3):
The difference in y-values is 3 - 2 = 1.
As we can see, the difference in y-values is not consistent between the pairs of x-values. In a linear function, the difference in y-values should be the same for all pairs of x-values.
Therefore, based on the table, the function is nonlinear.