The table shows a function. Is the function linear or nonlinear?
x y
7 2/5 4
8 1/5 3
9 1
In a linear function, the rate of change of y with respect to x is constant.
Let's calculate the rate of change between the points (7 2/5, 4) and (8 1/5, 3):
Change in y = 3 - 4 = -1
Change in x = 8 1/5 - 7 2/5 = 1/5
The rate of change is -1 / 1/5 = -5
Now, let's calculate the rate of change between the points (8 1/5, 3) and (9, 1):
Change in y = 1 - 3 = -2
Change in x = 9 - 8 1/5 = 4/5
The rate of change is -2 / 4/5 = -10/4 = -5/2 = -2.5
Since the rate of change is not constant, the function is nonlinear.
To determine whether the function is linear or nonlinear, we need to check if there is a constant rate of change between the x and y values.
Let's calculate the rate of change (slope) between the given points:
For the points (7 2/5, 4) and (8 1/5, 3):
Change in y = 3 - 4 = -1
Change in x = 8 1/5 - 7 2/5 = 1/5
Slope = Change in y / Change in x = -1 / 1/5 = -5
For the points (8 1/5, 3) and (9, 1):
Change in y = 1 - 3 = -2
Change in x = 9 - 8 1/5 = 4/5
Slope = Change in y / Change in x = -2 / 4/5 = -10/4 = -5/2
The slopes between the two pairs of points are different, indicating that the function does not have a constant rate of change. Therefore, the function is nonlinear.