Question

Is the function in this table linear or nonlinear and why?
The function is not linear because the rate of change is not constant.
The function is linear because the y-values are multiples of 50.
The function is linear because the x-values increase by a constant number.
The function is not linear because there are negative values in the y-values.

У

-100
1
-50
2

3
100
4
150

Answer 1

To determine whether the function represented in the table is linear or nonlinear, we need to check if there is a constant rate of change between the x-values and corresponding y-values.

Let's list the provided pairs of (x, y):

- (1, -100)
- (2, -50)
- (3, 100)
- (4, 150)

Now, let's calculate the changes in y (Δy) for consecutive x-values (Δx):

1. From (1, -100) to (2, -50):
- Δy = -50 - (-100) = 50
- Δx = 2 - 1 = 1
- Rate of change = Δy / Δx = 50 / 1 = 50

2. From (2, -50) to (3, 100):
- Δy = 100 - (-50) = 150
- Δx = 3 - 2 = 1
- Rate of change = Δy / Δx = 150 / 1 = 150

3. From (3, 100) to (4, 150):
- Δy = 150 - 100 = 50
- Δx = 4 - 3 = 1
- Rate of change = Δy / Δx = 50 / 1 = 50

Now, we can see that the rates of change are not constant:

- From points (1, -100) to (2, -50), the rate of change is 50.
- From points (2, -50) to (3, 100), the rate of change is 150.
- From points (3, 100) to (4, 150), the rate of change is 50.

Since the rate of change is not constant, we can conclude that **the function is nonlinear**.

Therefore, the correct reasoning is: "The function is not linear because the rate of change is not constant."